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1383 


UNITED STATES OF AMERICA. 
















































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FLAGS OF ALL COMMERCIAL NATIONS. 



J. B.LIPPINC07T & CO. PHILADELPHIA. 



















































































































































POCKET-BOOK 

OP 

MECHANICS 


AND 

ENGINEERING. 

CONTAINING 

A MEMORANDUM OF FACTS AND CONNECTION 

OP 

PRACTICE AND THEORY. 

BY 

JOHN W. NYSTROM, C.E. 



SEVENTEENTH EDITION REVISED AND 

WITH 

ORIGINAL AE 


PHILADELP 
J. B. LIPPINCOTT & CO. 


LONDON: 16 SOUTHAMPTON ST., COVENT GARDEN. 
1883. 







Entered according to Act of Congress, in the year 1872 by 
JOHN W. NYSTEOM, 

In the office of the Librarian of Congress, at Washington. 
Copyright, 1883, by JOHN W. NYSTEOM. 












PREFACE. 


Every Engineer should make his own Pocket-Book, as he pro¬ 
ceeds in study and practice, to suit his particular business. The 
present work has been accumulated in that way during the author’s 
professional career. It was originally not intended for publica¬ 
tion, but grew too large for the pocket in form of manuscript, 
which circumstance, combined with repeated requests to publish it, 
first placed it before the public in the year 1854. 

The author claims to have given a goodly share of original 
matter, and has spent much labor and money in experiments on 
subjects requiring elucidation. 

The authors consulted are distinguished experimenters, such as 
Dalton, on air and heat; Regnault, on steam; Kopp, on the ex¬ 
pansion of water; Morin, on friction and strength of materials; 
Joule, on the’ mechanical equivalent of heat; the Franklin Insti¬ 
tute, on the strength of iron and copper at different temperatures ; 
the Royal Technological Institute, Stockholm, on dynamics; and 
various others of equal authority; but these savans are not re¬ 
sponsible for the formulas and tables which are herein deduced 
from their experiments. 

The solution of mathematical formulas leads to powerful pre¬ 
sumptions in the revelation of physical laws, which could never be 
attained or realized from mere observation of facts in experiments 
and practice. All observation and contemplation which involves 
mind, involves theory, which is the foundation of our practice 
and progress. 

A knowledge of algebra is not necessary for the use of the for¬ 
mulas, and it is satisfactory to know that most engineers who are 
not versed in mathematics have acquired the very important habit 
of inserting numerical values for the corresponding letters, which 
they prefer to cumbrous written rules, which are impracticable in 
extensive problems. If all the formulas herein were explained in 
words, the book would exceed in volume Webster’s unabbrevi¬ 
ated Dictionary, and the matter would be only so much the more 
complicated. The algebraical formulas herein are solved into all 
their functions, ready to receive what is given and refund what is 
required. They not only tell what is to be done, but at a glance 
impress the mind with the complete operation. 

JOHN W. NY STROM. 


1010 Spruce street, Philadelphia. 







CONTENTS. 


TAGS 

Accelerated motion, . . 303 

Acceleration, force by, , 309 

Acids, for soldering, . . 475 

“ binary compounds, 471,472 

Acoustics,.417 

Acres, .... 32,475 

Actual total amount of work, 310 
Addition in Algebra, . . 13 

Adhesion on rails, . . . 125 

Adulteration of metals, . 329 

Aerostatic & aerodynamic 354 
Age, moon’s, .... 496 

Air, composition of, . . 472 

“ and beat, . . . 387 

“ moisture in, • . 356 

“ pumps, dimensions of, 407 

“ required for furnaces, . 443 
“ “ “ forges,, . 439 

“ weight and volume of, . 389 

Algebra,.13 

Algebraical signs, ... 12 

Alligation, .... 19 

Alloys,.332 

Almanac, astronomical, . 500 

“ 19th century, . 497 

Alphabets for heading, . 511 

Amalgamation of gold, . 482 
Amplitude, .... 510 

Analytical geometry, . . 142 

Anchors and cables, . 

Ancient measures, . • . ,40 

Angles by a two-foot rule, 36 
Aneroid "barometer, . . 363 

Animal strength, . . 264 

Annuities,.25 

Annular double cylinder, 413 
Apothecaries’ weights, . . 32 

Apparent time, . . . 504 

Area of circles, circumf., 64, 72 
“ “ plain figure, . . 56 

“ “ solids, . . . .59 

Arithmetic, new system, . 44 

Arithmetical progression, . 20 

Artificial horizon, . . 504 

Asphalt, composition of, . 474 
Assaying of gold and silver, 478 
Astronomy, .... 496 
Atmosphere, .... 353 

Atmospheric refraction, 132, 503 
Atomic weights, . . . 470 

Attraction of the earth, . 309 
“ “ sun and moon, 310 

Audibility of sound, . . 417 

Avoirdupois weight, . . 33 

Axes, number on steamboats, 436 
Axles and shafts, . . . 418 

Azimuth,.510 

Balls, piling of, . . . 21 

“ weight and capacity of, 281 


PAGE 

Barometer, graduation of, . 355 
“ heightditT. places, 368 
“ measure of n’t by, 361 
Barrel measure, weight of, 475 
“ capacity of, . . . 62 

Bar-iron, weight of, . . 281 

Beams, solid and compound, 279 

“ strongest from a log, 275 

Beam, walking, . . 53, 63 I 

Bear or burden on animals, 264 | 
Bells, ringing, . . . .417 

Belting, . ... 372 

“ strength of, . . 378 

Belts, leather, . . . 265 

Bends of pipes, flow of wat’r in, 342 
Billiard problem, . . 322 

Binary compounds, . . 471 

Birmingham wire gauge, 289 
Blast engines and furnaces, 443 
Blasting, glycerine, dynamite, 473 
Blast, warm and cold, . . 443 

Blowing off salt water, . 430 

Blowing machines, fans, .. 442 

Board measure, . . 426 

Bodies in collision, . . 322 

Boilers, steam, * . .* 422 

“ stand’rd horse-power, 424 
“ strength of, . . 424 

“ staying of, . . . 425 

“ thickness of iron in, 431 

“ weight of, . . . 431 

Boiling-point of liquids, . 383 

“ “ salt water, 431 

“ temp, water, barom., 367 
Bolts and nuts, . . . 352 

Bramah’s hydraulic press, 335 
Breast wheel, water, . . 347 

Brick,.475 

Bridges,.301 

Buckets, num. on steamboats, 437 
Bushel, .... 32,471 

Cables and anchors, . 

“ weight, stren’th, price. 271 
Calculating machine, . 71 

Caloric, units of, . . . 392 

“ meter in st’m-boilers, 423 
Candle, spei’m, .... 476 
Cannons, heavy artillery, 393 
Capacity, measure of, . . 33 

“ and weight of balls, 281 
“ of solids, . . 60 

“ and w’ht materials, 328 

Cask, capacity of, . . .62 

Cast-iron pillars, strength of, 269 

“ pipes, weight of, . 283 

“ produce of, . . 443 

“ girders, . . . 274 

Castings, weight of by pat’rns, 433 
Catenaria, .... 255, 259 


4 











Contents. 


5 


?- 

9 PAGE 

Cements, concretes, . . 474 

Centimeters and inches, . 38, 364 
Centre of gravity, . . . 324 

“ “ gyration, . . 314 

“ “ oscillation, . . 320 

“ “ percussion, . . 324 

Centrifugal force, . . . 318 

Chains, stren’th, weig’t, price, 271 
Chain, surveying, ... 32 

Chains for railroads, . . 280 

Chapman’s rule, area & solid., 114 
Characters, signification of, 12 
Charge of powder, . . 306, 393 

Charcoal, .... 333 

Charge in blast furnaces, . 443 
Chemistry, .... 470 

Chemical formulas, comp’ds, 472 
Chime of bells, ... 417 

Chimneys, height of, . . 423 

Chlorination of gold, . . 482 

Chronology of events, . . 499 

Circle, formulas for, . . 48 

“ to square a, . . .54 

Circumf. and area of circles, 64,72 
Circular saw, .... 264 
Clearance of teeth, . . 296 

Climate, mean temperature, 358 
Clock, sidereal, . . . 498 

Coal, consumption of, . . 405 

“ weight and bulk of, 328, 426 
Charcoal, from diff’ent woods, 276 
Coefficients of vessels, . 458 

Cogwheels, .... 294 
Cohesive strength, . . 270 

Coins, American and foreign, 34 
Cold, artificial, . . . . 386 

Collision of bodies, . . 322 

Colors, spectrum, . . . 476 

“ tempering steel. . 332 

Columns, air, water, etc., 344, 353 
Combination, ... 18 

Combustion of fuel, . 405, 427 
Compass, mariner’s, . . 130 

Composition of air and water, 472 

Compounds, binary, chem’al, 471 
Compound interest, . . 24 

Compound steam-engines, 413 
Compressive strength, . 268, 474 

Concave & conv. mirr’s & lens, 486 

Concretes,.470 

Cone pulleys, .... 376 

Condensing water, . . . 408 

Confused terms in dynamics, 310 
Conducting p’r heat, elec’y,386,477 
Conic sections, . . . 142 

Construction of ships, . . 444 

Consumption of fuel, . 405 

“ “ gas, . . 372 

“ “ water, . 342 

Copper, strength at high heat, 433 
Cord of wood, . . . 426 

Cosine, cotang’t, cosecant, 163-253 
Couplings for pipes, plum’ng, 433 
Crank and pin, . . . 298 


PAGE 

Creation of the world, . . 498 

Cube and cube root, . . 82 

Cubic contents, ... 60 

Cubic inches water, iron, lead, 281 

Cupola,.435 

Curvature of the earth, 131, 132 
Curves for railways, 116-121 

Cut-off steam, expansion, 402 
Cut-off valve, ..... 420 
Cycle of the sun and lunar, 498 
Cycloid, .... 47, 307 

Day and night, length of, 510 
Dates, civil, astronm., mar’s, 498 
Decimal fractions to vulgar, 36 
Decimals of an hour, degree, 208 
Deaf and dumb alphabet, 511 
Declination of the sun, . 500 
Degree of the earth’s circle, 32 
Departure, . . . 128 

Dew point, .... 356 

Diagram, indicator, Plate V., 421 
Diamond, .... 35 

Diameter of wro’t-iron shafts, 418 
Difference in longitude, . 291 

Differential calculus, . . 26 

Dip of horizon, . . . 131 

Discount or rebate, . . 17 

Displacement of vessels, 444, 457 
Distances on the Amer. coast, 492 
“ U. S. railways, . 495 

“ in the world,. . 494 

“ in Europe, . 493 

“ of objects at sea, . 131 
“ spherical, . . 138 

Distance to the sun and moon, 292 
Distillation of coal oil, . 383 

Division in algebra, . . 14 

Divergence of the parallel, 133 
Dodecahedron, ... 55 

Double cylinder expansion, 413 
Drain, motion of water in, . 339 
Dredging machine, . . 265 

Duodenal system, ... 44 

Dynamics, . . 262,310, 314 

Dynamical terms, . 262, 310, 372 
Dynamite, glycerine powder, 473 
Barth, dimension of, . 488 

“ attraction of and on, 309, 310 
Eccentrics, .... 420 

Eclipses of Jup’r’s satellites, 505 
Economy of expan. of steam, 404 
Effect, dynamic, , . . 262, 311 

“ of steam-engines, 412 
“ of water, natural. . 344 
Electricity, positive and neg., 477 
Elevations above sea, . 367, 370 
Elementary substances, 470, 477 
Ellipse, construction of, . 46 

“ periphery of, . . 54 

“ formulas for, . . 144 

Elliptic mirrors, . . . 484 

“ sterns of vessels, . 456 
Ellipsoid, . . . 61 

Elasticity of beams, . . 273 








6 


Contents. 


PAGE 

Elevation of external rail, 119 
Embankments, . . . 122 

Engineer's command, . 431 

Engines, steam, of dif. kinds, 413 
Epact of the year and month, 496 
Equation of time, . . . 500 

Equivalents, atom weights, 470 
Evaporation on seas, . . 360 

Events before and after Christ,499 
Evolute of a circle, . . 47 

Expansion of air by heat, 387 

“ of water “ . 394 

“ of bodies “ . 384, 409 

“ of steam “ . 402 

Explo. nitro-glycer., dynam., 473 
“ of steam-boilers, . 436 

Excavation and embank., 122 

Eyes, long and near-sighted, 483 
Faces of the moon, . . 508 

Falling bodies, table for, . 308 

Fan, or ventilator, . . 442 

Fathom,.32 

Feed pumps or force pumps, 406 
Feet per sec. = miles per hr., 352 
Feet and metres, . 38, 40,364 

Fellowship, .... 17 

Felt covering for steam-pipes, 428 
Fine, and val. of sil. and gold, 480 

Fire-engines, . . . 337 

Fixed stai’s, .... 506 
Flags of nations—plate, . 1 

Flour-mills, .... 264 
Flooring, beams in, . . 279 

Flow of water in pipes, . 342 

“ in rivers, . 340 

Flues for steam-boilers, . 423 
Fly-wheels, .... 315 

Focal distance of lenses, . 486 
Foot-valves in air pumps, 408 

Force defined, . . 262,311 

Force, quantity of, moving, 310 
Force pumps, .... 407 
Force of temperature, . 385 

Forging by steam-hammers, 303 
Foreign money, . . 34 

“ weights and measures, 4o 
Fractions, vulgar into deeds, 36 
Freezing mixture, . 383, 386 

French metrical system, . 37 

Freight or load on R. R., . 124 

Friction, .... 266, 373 

Fuel, properties of, . . 426 

“ consumption of, . 405 

Fulcrum,.254 

Funicular machine, . . 259 

Fusion, temperature of, 332, 383 
Gallons^ standard, . . 32 

Gas, motion of, in pipes, . 476 
Gauge, American wire, . 288 

“ Birmingham, . 289, 475 

“ railway, ... 126 

“ for sheet, nails, rivets, 432 
Gearing, construction of, . 294 

Geography, . . . .488 


PAGB 

Geometry, .... 45 

“ analytical, . . 142 

Geometrical progression, . 22 

“ scale in music, 374 

Giffard’s injector, . . 439 

Girders, compound, iron, . 279 
“ cast iron, . . 274 

Glass, window, . . . 290 

Glues,.471 

Gold metal, .... 332 
Gold and silver, . . 478-480 

Gold, sil., platin’m, weight of, 479 
Golden number, . . . 498 

Governor, .... 319 
Gravitation, .... 304 

Gravity, specific, . . . 328 

“ centre of, . . 324 

Gunpowder, properties of, 393 

Guns, heavy artillery, . 393 

Gyration, centre of, . 314, 317 
Hammers, steam, . . 303 

Half-trunk steam-engine, 413 

Hardness of substances, 276, 333 
Heat, caloric, .... 379 
“ as a mode of motion, 310 

“ latent, .... 382 
“ lost by radiation, . 428 

“ specific, . . .390 

“ units and horse-power, 392 
Heating surface in boilers, . 421 
“ houses by steam, 429 

Helix of a screw, construct. 53, 54 
Hexahedron, ... 55 

Hemp ropes, weight and price,271 
High water, time of, . . 508 

Height of the a tmosphere, 351, 361 
“ measure of by barom., 361 
“ of cities, . . 358, 370 

“ of snow-line, . 353 

“ mountains and vole’s, 363 
“ miscellaneous, . 370 

“ col. water, air, mer., 344, 353 
Hodgkinson’s strength pil’s., 269 
Horizon, artificial, „ . . 504 

Horizontal range, . . 306 

Horse-mill, . . . 264,341 

Horse-power, . . 262,410 

of engines, 414, 416 
of boilers, stan., 424 
“ of locomotives, 122 
Horse, ability of, . . 125 

Hose, velocity of water in, . 337 
Humidity in the air, . 356 

Hydraulics, .... 336 
Hydraulic mortar, . . 474 

“ press, . . .355 

“ ram, . . 343 

radius in rivers, 340 
“ pumping water, 341 
Hydrodynamics, . . .344 

Hydrometer, ... 334 

Hygrometry, .... 356 
Hyperbolic mirror, . . 484 

Hyperbolic logarithms, . 143 





Contents. 


7 


PAGE 

Hyperbola, equation for, . 145 

Icosahedron, .... 55 

Ice, expansion, contraction of, 384 
Inches to decimals of a foot, 36 
Inches and centimetres, 38, 364 
Inclined plane, . . . 254, 260 

Indicator diagram, Plate Y. 421 
Index of refraction, . . 485 

Inei'tia,.310 

Injection water, . . .408 

Injector, Gilfard’s, . . 439 

Incrustation in boilers, . 430 

Integi’al calculus, . . 28 

Interest, sim., compound, 16,24 
Interpolation, ... 30 

Impact of bodies, . . . 323 

Iron or blast furnaces, . 443 

“ acid test for quality, . 418 
“ strength of at high tern., 433 
Irrigation, vol. of water for, 359 
Joints, proportion of riveted, 425 
Jonval’s turbine,. . . 348 

Jupiter’s satellites, . 505,508 
Joule’s equivalent of heat, 392 
Julian period, . . . 498 

Lakes, area of, 360 

Land surveying, . . . 128 

Lap and lead on slide-valves, 420 

Latent heat,.382 

Lateral strength, . 272,278 

Latitude and longitude, . 490 

Law of gravity, . . . 303 

Leap year,.498 

Leather belts, . . 265,378 

Lenses optical, . . . 486 

Letter for printing, . . 511 

Lever, static momentum, 254, 256 
Light and color, . . . 372 

Liquid measure, ... 33 

Llama of Peru, ... 264 

Load on roofs, .... 299 
Locomotive, traction, . 124 

Logarithms of numbers, . 148 

“ trigonometric, 163 

“ hyperbolic, . 143 

Longitude of places fr. Green., 490 
“ difference in time, 491 

Log-line, length of, . . 32 

Lunar cycle, . . . . 498 

Magnifying power of lenses, 486 
“ opera-glasses & teles’p’s, 487 
Mass, explained, . . . 309 

Mantissa of logarithms, . 146 

Manual labor, .... 264 
Mariner’s compass, . . 130 

Mariner’s date, .... 498 
Maxima and minima, . 30 

Mean time, .... 498 
Measures, ancient, . . 40 

“ fox-eign, ... 40 

“ and weights, . 32 

Mechanics, .... 254 
Men’s power, .... 262 

Meridian, to find the, . . 509 


PAGE 

Meta-centrum of ships, . 446 

Metals, hardness of, . . 333 

Metre and feet, . . 38, 40, 364 

Metrical system, ... 37 

Microscope, .... 486 

Mile, statute and nautical, . 32 

Miles pr. hour = ft. pr. second, 352 
Miles and kilometers, . . 39 

Mills, flour, saw, rolling, . 264 

Minerals, composition of, . 471 
“ hardness of, . . 333 

Mirrors, convex and concave, 484 
Momentum, static, . . . 254 

“ dynamic, . 310 

“ in bodies, . . 322 

Moment of inertia, . . 310 

Money, American & foreign, 34 
Moon, elements of, . . 496 

Moon’s faces, .... 508 
Mortar and cements, . . 474 

“ piece of ordnance, 307, 393 
Motion of bodies in collision, 322 
“ quantity of, mode of, 310 
•“ of water in pipes, . 342 
“ of water in rivers, 340 
“ of gas in pipes, . 476 
Mountains, height of, . 363 

Monuments, height of, . . 370 

Multiplication in algebra, 13 
Music, acoustics, . . . 417 

Natural sines, etc., . . 209 

Nails, penny, length & wei’ht, 432 
Navigation, traverse, . . 128 

Night and day, length of, . 508 
Nitro-glycerine, dynamite, 473 
Nominal horse-power, . . 410 

North and South, to find, . 509 

Notation of numbers, . . 11 

Nuts and bolts, size & weight, 290 
Nystrom’s calculator, . . 71 

Octahedron, ... 55 

Obstructions in rivers, . . 341 

Opera-glasses, ... 487 

Optics,.483 

Ordnance, heavy, . . . 393 

Oscillation of pendulum, . 321 
Overshot-wheel, . . . 347 

Paper, drawing and tracing, 352 
7T, value of, to 45 decimals, . 48 

Parabola, to construct a, . 47, 145 
Parabolic construe, of ships, 444 
“ mirror, . . . 484 

“ vein, . . . .339 

Paradox, hydraulic, . . 335 

Parallax, sun’s, . . .503 

Pattern-makers’ rule, . 433 

Peal of bells, . . . .417 

Pendulum, .... 320 

Penny nails, .... 432 
Perch of stone, . . . 475 

Percussion, centre of, . . 324 

Periphery of circles, . . 64, 72 

“ “ an ellipse,. . 54 

Performance of steam-ships, 458 






8 


Contents. 


PAGB 

Permutation, .... 18 

Piling ot balls and shells, 21, 23 
Pile-driving, . . . .277 

Pipes, cast-iron, weight of,. 283 
“ of diffnt metals, w’t of, 282 
“ brass and copper, . 432 
“ motion of water in, 337, 342 
“ and flues, . . . 423 

“ motion of gas in, . 476 

“ steam, radiat. of heat, 429 
“ steam, size of, . . 409 

Pitch of propellers, . . 467 

“ “ screw, ... 53 

“ “ spirals, . . 47,54 

“ “ screw thread, . 290 

“ “ teeth in gearing, . 296 

Planetary syst’m, el’ments of, 506 
Plane, inclined, . . . 260 

“ sailing, traverse, . 127 

Platinum, weight of, . . 479 

Plumbing, . . . 432 

Points of the compass, . . 130 

Polygons.63 

Polyhedrons, .... 55 

Poncelet’s water-wheel, . 346 

Population countries & cities, 489 
Portland cement, . . 474 

Ports, steam.408 

“ high water in, . 506 

Powder, properties of, . . 393 

Power, actual horse, . . 411 

“ nominal “ . . 410 

“ of steam-engines, . 413 

“ “ locomotives, . . 125 

“ “ a man, horse, 262, 261 

“ in moving bodies, . 313 
“ definition of, . 262,311 

“ for different mills, . 264 

“ “ pumping, . . 341 

“ “ punching, . . 435 

“ “ steamboats, . 460 

“ magnifying, . . 486 

“ reflecting of heat, . 386 
“ for blowing machines, 440 

“ “ fans, ventilation, 442 

“ “ quartz mills, . 482 

Pound, avoirdupois, troy, . 33 

Prime, vertical and parallel, 133 

Press, hydraulic, . . 335 

Pressure col’m’s wat’r, etc., 314,351 
Price ol' boiler tubes, . . 429 

“ “ copper & brass tubes, 432 

“ “ couplings for plum’g, 432 

“ “ gold and silver, . 480 

“ “ hemp and wire x’opes, 271 

“ “ rolled iron, diff. f’m’s, 280 

'* “ taps, dies, stocks, . 432 

“ “ turbines, . . 350 

“ wrouglit-irongird’rs, 279 
Projectiles for guns, . 306,393 

Propellers, screw, . . . 466 

Proportion in arithmetics, 15 

Pulleys.258 

Pumps, air, force, . . 407 


PAGE 

Pumping, water, . . . 341 

Punching iron plates, . 435 

Pyrites, iron, gold-bearing, . 482 
(Quantity defined, . . H 

“ of motion, . . 310 

“ “ work, total, . 310 

“ “ moving force, 310 

Quartz mills, . . . . 482 

Radiation of heat, . 386, 428 

Radius of the earth, . 309, 488 
Roads, bad, in Peru, . . 264 

“ traction on, . . 126 

Rail, elevation of outer, . 119 

Railroads, .... 116-126 
Rails, spring of, . . 119 

“ weigh t and price of, . 280 
Railway curves, ,. . . 116 

Ram, hydraulic, . . . 343 

“ in pile-driving, . 277 

Rain, fall of, . . . 126, 359 

Range of a projectile, . 307 

Rebate or discount, . . 17 

Rectang’r beams from a log, 275 
Reduction of inches to feet, 36 
Reflecting power of heat, . 386 

Refraction, light, atmos., 132, 503 
Refractivb indexes, . . 485 

Reg’ng a time-keeper by stars, 507 
Resistance and force of wind, 352 
“' ‘ in water, . . 341 

“ in -railway curves, 352 

“ of air to projectiles, 306 

Resultant of forces, . . 257 

Retarded motion, . . . 313 

Right ascension, sun’s, . 500 

Ringing bells, .... 417 
Rivers, length of, . . 359 

“ flow of water in, . 340 
“ obstruction in, . 341 

Riveted joints, proportion of, 425 
Rivets, iron and copper, . 433 

Roasting of sulphurets, . 482 
Roebling’s wire ropes, . 271 

Roll’d i ron, weight, size, price, 280 
Rolling-mills, . . 264,314 

Roman cement, . . . 474 

“ notation, . . 12 

Roofs, wood and iron, . . 299 

Ropes, strength, weight, size, 271 

Roots, square and cube, . 82 

Rule for pattern-makers, . 433 

Russia sheet iron, . . . 432 

Safety-valve, . . . 408 

Sailing distances, . . . 492 

Salt water in boilers, incrus., 430 
Sash-bars, iron, for windows, 280 
Satellites, Jupiter’s, . . 508 

Saturation in boilers, . . 430 

Saw-mills, circ. & alternative, 264 
water-wheel, . 347 

Scale, decimal inch, . . 296 

Screw jack, force by, . . 261 

“ helix, . . . .53 

“ propeller, . . .' 466 


J 












Contents. 


9 


PAGE 

Screw threads, . . . .290 

Seasons,.358 

Secant, natural, . . .200 

Segments of a circle, . . 78 

Setting of stars, . . .507 

Shafts, diameter, revolutions, 418 
Sheering iron plates, . . 435 

Sheet, iron, copper, etc. 287, 289, 443 
Ships, construction of, . . 446 

Shrinkage of castings, . 433 

Sidereal time, clock, year, . 498 
“ and solar times, . 503 

Sidings of parallel tracks, . 121 
Signs, signification of, . 12 

Silver, to refine, . . .481 

“ and gold table, value, 480 
Silvering metals, . . . 475 

Simple interest, ... 16 

Simple substances, . . 470, 477 

Simpson’s rule, ... 114 

Sines, cosines, etc., natural, 209 
“ “ logarithmic, 163 

Slide valves, .... 420 
Slip of propellers, . . 466 

Slope of embankments, . 122 
Smelting or freezing-point, 383 
Snow-line and bulk of snow, 353 
Snow, weight of new-fallen, 299 
Solar time to sidereal, . . 503 

Solders for bracing, . . 332 

Soldering, acids for, . . 475 

Solidity of revolution, . 114 

Solids, capacity of, . . 60 

Sound, velocity of, . . 417 

Soundings to low water, . 509 
Specific gravity, . . . 328 

“ heat, caloric, . . 390 

Spectrum, colors, . . 476 

Speed & horse-power of steam.,460 
Spheres, sur., capac., weight, 281 
Spherical distances, . . 138 

“ mirrors, . . 484 

“ trigonometry, . 138 

Spiral, construction of, . 47, 54 

Spindle, circular, ... 62 

Spring of rails, ... 121 

Square a circle, to, . . . 54 

“ and square roots, . 82 

Standard horse-pow’r of boil., 422 
Stability of ships, momentum, 469 
Static momentum, . 554, 311 
Stars, R. A. and declination, 506 
“ setting of, 507 

Steam, properties of, . . 394 

“ boilers, .... 422 

“ condenser, . . 411 

“ engines, . . . 413 

“ expansion of, . . 402 

“ hammers, . . . 303 

“ loss by rad. from pipes, 428 

“ pipes and ports, . 408 

“ superheated, . . 438 

“ ship performance, . 460 

“ table, . . . 400, 410 


PAGE 

Steel, tempering of, . . 332 

Stocks and dies, price of, . 432 

Stone, perch of, ... 471 
Strength of materials, 268-279 
“ iron, cop., high heat, 433 

“ boiler flues, . 422,437 

“ Portland cement, . 474 

“ of animals, . . 264 

“ of belting, . . 378 

Subtraction in algebra, . 13 

Sulphurets, gold-bearing, . 482 
Sun, set and rise, . . 510 

“ cycle of the, . . . 498 

“ distance to, . . . 310 

“ parallax of, 503 

“ properties of, . . 496 

“ R. A. and declination, 500 

Superheating steam, . . 438 

Surface of pulleys, . . . 373 

“ of boiler tubes, 429,432 
“ of revolution, . . 114 

“ of solids, ... 60 

Surveying, .... 89 

“ chain, ... 32 

Suspension bridges, . . 302 

Tangent, etc., natural, . 209 
“ logarithms, . 163 

Taps and dies, .... 432 
Teeth for wheel, gearing, . 294 

Telescope, astronomical, . 487 
Temperature, mean, climate, 358 
“ color of steel, 332 

“ force of, . 385 

“ fusion, alloys, 332 

" miscellaneous, 386 

“ correction for, 366 

“ on the ocean, 370 

“ boiling, smelt., 383 

“ table, conver’n, 380 

“ boil, water bar., 367 

Terms, dyn’cal, proper, conf., 310 
Tests, chemical, for metals, 481 
Tetrahedron, elements of, 55 
Thickness of boiler-iron, . 434 
Thermometer, graduation of, 379 
Threads, screw, num. pr. inch, 290 
Threshing machine, . . 264 

Timber, green and seasoned. 426 
Time chronology, . . 498 

“ apparent, . . . 504 

“ equation of, . . 500 

“ sidereal, . . .> . 503 

“ diff. longitude, . . 491 

“ to regulate a watch, 507 
“ of high water, . * 508 

Tinder, temperat’re of firing, 388 
Tin-plates, English, . . 352 

Tinning, acids for, . . . 475 

Tonal system or duodenal, 44 
Tone of bells, .... 417 
Tonnage register, Amer., Eng, 465 
Tracing paper, .... 352 
Traction on roads, . . 125 

Traveling distances,. . 492-495" 


L 















10 


Contents. 


PAGE 

Traverse surveying and table, 90 
Triangles, plane, . . . 136 

“ spherical, . . 139 

Trigonometry, plane, . . 134 

“ spherical, . 138 

Tropical year, .... 498 

Troy weight.33 

Truss-bridges, .... 303 
Tubes, iron, lap welded, . 429 

“ copper and brass, . 432 
Turbines, Jonval’s, . . 348 

Tuyeres in furnaces, . . 440 

Type metal, .... 332 

“ proportions of, . . 371 

Undersliot water-wheel, . 346 

Units of work and heat, . 392 
“ of power, dynamics, 262 
“ workman-days, . . 262 

U.S. stan. weight and meas., 32 
U. S. tonnage law, . . . 463 

Valves, air-pump, . . 407 

“ safety, .... 408 
“ slide, . . . 420 

“ blast-engines, . . 440 

Vegetable acids and salts, 472 
Vein of water, contracted, . 339 
Ventilator, fan, . . . 442 

Vessels, tonnage of, . . 463 

“ construction of, . 446 

Velocity of light and electr’y, 476 
“ of sound, . . 476 

“ of water in pipes, 342 

“ of water in rivers, 340 
“ of falling bodies, . 308 
“ of projectiles, . 393 

“ virtual, . . . 310 

Vis-viva, principles of, . 310 

Vibration of pendulum, . 320 
“ in musical tones, 417 
Volcanoes, active, height of, 363 
Vulgar fractions to decimals, 36 
Walking-beam, . . 53, 63 

Warren girder, bridges, . 301 

Water, properties of, . . 394 

“ composition of, . 472 


PAGK 

Water colors, . . . 351 

“ motion of in pipes, 342 
“ “ “ rivers, 340 

Water-wheels, . . . 346 

“ injector, . . . 439 

“ works, . . . .341 

“ falls, .... 369 

Wave-line, proper’s of waves, 449 
Weather, prediction of, . 355 

Wedge, .... 255, 261 
Weight and measure, . 32 

“ “ new system, 44 

“ and capacity of balls, 281 
“ round & square iron, 282 
“ of pipes, . . . 282 

“ cast-iron, cylin., pip’s,283 
“ flat rolled iron, . 284 

“ copper bolts, . . 287 

“ pr sq.ft, sheet, 287-289,299,433 
“ and capacity, . . 291 

“ castings, . . . 433 

“ steam-hammers, . 303 

“ & bulk of substances, 328 

“ engines and boilers, 434 
“ bells, ... 417 

“ cub. in. water, etc., 281 
“ heavy ordnance, . 393 

Weirs, water flowing over, 340 

Wheels, water, ... 346 

Wind, force by, . . . 354 

“ mills, .... 354 

“ velocity of, . . . 354 

Window-sashes, iron, . 280 

“ glass, . . . .352 

Wire-gauges, ... 288 

“ ropes, Roebling’s, . 271 
Wood, for combustion, cord, 426 

“ South American, . 276 

Work, dynamic, 262, 311, 314, 392 
“ actual total amount, 310 
“ total quantity of, . 310 

“ rate of, .... 310 

Yard, feet and inches, . 32 

Years, different kinds of, . 498 
Zinc, sheet, weight of, 287, 433 












Mathematics. 


11 


INTRODUCTION. 

Quantity is that which can he increased or diminished by augments or 
abatements of homogeneous parts. Quantities are of two essential kinds. 
Geometrical and Physical. 

1st, Geometrical quantities are those which occupy space; as lines, surfaces, 
solids, liquids, gases, &c. 

2nd, Physical quantities are those which exist in the time hut occupy no space, 
they are known hy their character and action upon geometrical quantities; as 
attraction, light, heat, electricity and magnetism, colours, force, power, &c., &c. 

To obtain the magnitude of a quantity we compare it with a part of the same, 
this part is imprinted in our mind as a unit, hy which the whole is measured 
and conceived. No quantity can be measured hy a quantity of another kind, 
hut any quantity can be compared with any other quantity, and by such com¬ 
parison arises what we call calculation or Mathematics. 

-- 

MATHEMATICS. 

Mathematics is a science by which the comparative value of quantities 
are investigated; it is divided into : 

1st, Arithmetic* —that branch of Mathematics, which treats of the nature 
and property of numbers; it is subdivided into Addition, Subtraction, Multipliear 
tion, Division, Involution, Evolution and Logarithms. 

2nd, Algebra*—that branch of Mathematics which employs letters to repre¬ 
sent quantities, and by that means performs solutions without knowing or 
noticing the value of the quantities. The subdivisions of Algebra are the same 
as in Arithmetic. 

3rd, Geometry* —that branch of Mathematics which investigates the rela¬ 
tive property of quantities that occupies space; its subdivisions are Longimetry, 
Planemetry, Stereometry, Trigonometry, and Conic Sections. 

4th, Differential-calculus, — that branch of Mathematics, which ascer¬ 
tains the mean effect, produced by group of continued variable causes. 

5th, Integral-calculus, —the contrary of Differential, or that branch of 
Mathematics which investigates the nature of a continued variable cause, that 
has produced a known effect. 

- *4 - 

ARITHMETIC. 

The art of maneuvering numbers, and to investigate the relationship of 
quantities. 

Figures —1, 2, 3, 4, 5, fl, 7, 8, 9. Arabic digits, about nine hundred years old. 

Ciphers — 0 , 0 , 0. Sometimes called noughts, it is the beginning of figures and 
things. 

Number is the expression of one or more figures and ciphers. 

Integer is a whole number or unit. 

Fraction is a part of a number or unit. 

When figures are joined together in a number, the relative dignity expressed 
l y each figure, depends upon its position to the others. Thus, 



674,385 ; 496,345 ; 695,216 ; 505,310 : 685 , 3 6 7 ; 













1 


Notation. 


ROMAN NOTATION. 

The Romans expressed their numbers by various repetitions and combinations 
of seven letters in the alphabet; as, 


1 = 1 . 

2 = 11 . 

3 = III. 

4 = IV. 

5 = V. 

6 = VI. 

7 = VII. 

8 = VIII. 

9 = IX. 

10 = x. 

20 = XX. 

30 = XXX. 

40 = XL. 

50 = L. 

60= LX. 

70 = LXX. 

80 = LX XX. 

90 = XC. . 

100 = C. _ 

Examples.— 1872.—MDCCCLXXII. 524,365, DXX1VCCCLXV. 

An imperfection in the Roman Notation consists in the fact that there is no sig¬ 
nification for the cipher, as in the Arabic Notation. 


500 = D, or LO. 

1,000 = M,'or CO. 

2,000 = MM, or HOOD. 

5,000 = V, or LOO. 

6,000 = VI, or MMM 
10,000 = X, or COO. 

50,000= L, or LOOO. 

60,000 = EX, or MMMO. 

100,000 = C, or COOO. 

1,000,000 = M, or COOOO. 

2,000,000 = MM, or MMOOO. 

A bar, thus, — over any number, in¬ 
creases it 1000 times. 


Signification' of Characters. 

= Equality, as 6 = 6, reads 6 is 
equal to 6. 

+ Plus , Addition, . 3 -f 6 = 9 
— Minus, Subtraction, 6 — 2 = 4 
X Multiplication, . 3 X 4 = 12 
-s-or: Division, . .15:5 = 3 
y Square root, . . . j/ 9 = 3 
#"8 = 2 
8>4 


Cube root 
Greater, 

Less,.6 <^9 

oo Infinite,. -g 

Astronomical Characters. 

Planets. 

© The Sun. 
d The Moon. 

$ Mercury, 

9 Venus, 

© The Earth. 
f Mars. 

5 Ceres. 

$ Pallas. 

$ Juno. 

§ Vesta. 

If. Jupiter. 
ll Saturn. 

Ijl Uranus. 

Neptune. 


/ Integral, . . . fdy = y. 
dy Differential, . . dy = dx - f-. 

3/4 Fraction. = $. 

Ship sign, dead flat, 

I I Furnace fire-grate. 

O Boiler heating-surface. 

% Sharp. High. 

]y Flat. Low. 

7r Periphery. 


(3 Conjunction in the 
same degree or sign, or 
having the same longi¬ 
tude or Right Ascension. 

•)f Sextile, when two signs 
distant, or differing 60° 
in longitude or Right 
Ascension. 

□ Quartile, when three 
signs distant, or differ¬ 
ing 90° in Longitude or 
Right Ascension. 

S Opposition, when six 
signs distant, or differ¬ 
ing 180° in Longitude 
or Right Ascension. 

^ Ascending Node. 

£3 Descending Node. 

R. A. Right Ascension. 


Signs of the Zodiac. 

t Aries > • • §4 

8 Taurus, . . pjy 
El Gemini, . . f|v| 

zz Cancer, . . 

£1 Leo, . . . 

TIE Virgo, . . . g; 
— Eibra, . . ^ 

m Scorpio, . . 

$ Sagittarius, 

Vf Capricornus, 
Aquarius, . 

X Pisces, . . 


















Algebra. 


13 


- | 

ALGEBRA. ! 

! 

In Algebra we employ certain characters or letters to represent quantities. 
These characters are separated by signs, which describe the operations ; and by 
that means, simplify the solution. 

1. Whatever the value of any quantity may be, it can be represented by a 
character, as a. Another quantity of the same kind, but of different value, be¬ 
ing represented by b. The sum of these two quant ities is of the same kind but 
of different value. 

For Addit ion we have the algebraical sign -f, (plus) which, when placed 
between quantities, denotes they shall be added; as a-f6, reads in the 
algebraical language, “ a plus 6,” or a is to be added to b. 

Another algebraical sign =, (Equal) denotes that quantities which are placed 
on each side of this sign, are equal. Let the sum of a and b be denoted by the 
letter c; then we have, 

a-\-b=c. 

This composition is called an algebraical equation. The quantity on each side 
of the equal sign is called a member , as a-\-b, is one member, and c, the other. When 
one of the members contains only one quantity, that member is generally 
placed on the first side of the equal sign, and its value commonly unknown; 
but the value of the quantities in the other member being given, as a=4, and 
6=5, then the practical mode, to insert numerical values in algebraical equar 
tions, will appear; as, 

Equation, c=a+6, 

4-f 5=9, the value of c. 

2. The sum of three quantities a, b, and c, is equal to d, then 

Equation, d=a-\-b-\-c, 

4-}-5-j-9=18, the value of d. 

3. For Subtraction we have the algebraical sign,—, (minus) which, when 
placed before a quantity, denotes it is to be subtracted as, a — b, reads in the 
algebraical language “a minus 6,” or from a, subtract b. Let the difference be 
denoted by the letter c; and a=8. 6=3 

Equation, c=a —6, 

8—3=5, the value of c. 

4 . From the sum of a and 6, subtract c, and the result will be d ; then, 

Equation, d=a-\-b —c, 

8-j-3—5=6, the value of d . 

5. When two equal quantities are to be added, as a-\-a. it is the same as to 
take one of them twice, and is marked thus 2a. The number 2 is called the 
coefficient of the quantity a. If there are more than two equal quantities to be 
added, the coefficient denotes how many there are of them; as, 

Equation, - - - - a-|-a=2a, B 

“ a-fa-j-a=3a, 

“ a-pa-i-a-j-a=4a, 

die., tfc. 

When the quantities are separated by the signs, plus, or minus, they are 
called terms. 

6. Multiplication.—When a quantity a, is to be multiplied by another 
quantity 6, then a and 6 are called factors ; and separated by no sign as ab; 
which denotes that a is to be multiplied by 6 ; but when the values of a and 6 
are expressed by numbers, they are separated by the sign X (Multiplication); the 
result from Multiplication is called the product. Let a=8, and 6=6, and the pro¬ 
duct of a and 6, to be c, then, 

Equation, c=ab, 

8X6=48, the value of c. 

7. The product of a and 6, is to be multiplied by c, and the latter product will 
be equal to d ; then, 

Equation, d=abc, 

8X6X^8=2304, the value of d. 








14 


Algebra. 


8. The sum of a and 6, is to be multiplied by c, and the product will be d; 
then, 

Equation, d = c (a+6), 

48 (8 + 6) = 672 the value of d. 

When the sum of two or more quantities is to be multiplied by another quan¬ 
tity, the sum is to be enclosed in parentheses, and denotes itself to be one factor. 
The other factor is to be placed on the outside of the parentheses, as seen in the 
preceding example. 

9. To the product of a and c, add b, and the result will be d; then, 

Equation, d= ac +6, 

8X48+6 = 390 the value of d. 

Be particular to distinguish the two Examples 8, and 9. 

10. The sum of a and b, to be multiplied by the sum of a and c ; the product 
will be d; then, 

Equation, d— (a +6) (a + c), 

(8+6) (8+48) = 784. 

11. The sum of c and b, to be multiplied by the difference of c and a ; the re¬ 
sult will be d ; then, 


Equation, d — ( c+ b) (c —a), 
(48+6) (48—8) = 


2160. 


12. Division. —When a quantity a, is to be separated into b equal parts, the 
numbers of parts or b, is called the divisor, and the value of each part, is called 
the quotient. The sum of the parts or the whole quantity a, is called the dividend; 
a and b, is separated by the sign : (Division); as a : b, reads in the algebraical 
language, “a divided by 6.” Let the quotient be denoted by the letter c; and 
a=18, 6=6, then, 

Equation, c — a: b, 

18 : 6 = 3 the quotient c. 

In Algebra it is found more convenient to set up Division as a fraction, then 
j it will appear as, 

13. Divide a, by c, and the quotient will be 6. Then, 

a 

Equation, b = 

18 

-g- = 6 the quotient b. 

14. The product of a and 6, to be divided by c; and the product will be d. 
Then, 

_ , a6 

Equation, d =-’ 


18X6 36 

— = 36. 

15. The sum of d and b, to be multiplied by c, and the product divided by a; j 
then the result will be e. 


Equation, e — 


c ( d+b ) 


a 

3 (36+6) 


18 


7. 


16. From the product of a and c, subtract 36; divide the remainder by the 
difference of a, and c; the result will be h. 












Proportion. 


1ft 


et .. , ac—ob 

Equation, h — -, 

a — c. 


18X3—3X6 

18—3 


= 2 . 4 . 


An old man said to a smart boy, “ How old are you ?” to which he replied.— 
“To seven times my father’s age add yours, divide the sum hy double the 
difference of yours and his, and the result will be my age.” 

Letters will denote, 
a = the old man’s age, 
b — the father’s age, 
c = the boy’s age. Then, 


Equation, c ■ 


lb-\-a 


2 (a—6) 


the boy’s age. 


Now for any number of years of the old man and the father, will be a cones* 
ponding age of the boy; suppose, 

a = 73 years the age of the old man, 
b = 57 years the father’s age. 

Require the boy’s age. 


c — 


7X57+73 
2 (73—57) 


=14$ years. 




PROPORTION. 

The relative, value of two quantities, is obtained by dividing one into the other, 
and the quotient is called the ratio of their relationship. If the ratio of two 
quantities is equal to the ratio of two other quantities, they are said to be in the 
same proportion; as, 

a: b = c: d, 

reads in th.e algebraical language “ a is to b as c is to dr—a, b, c, and d, are call¬ 
ed terms, of which a is the first, b the second, c the third, and d the fourth term. 
The first and fourth are called “ the outer terms,” and the second and third, 
“ the inner terms.” The whole is called an “ analogy.” 

A property in the nature of analogies is, that the product of the outer terms 
j ’ to the product of the inner be. Suppose a = 4, b = 9, c = 12, 


d — 27 . 


4: 9=12 : 27, 
ad=bc, 4X27=9X12. 


If any one of the four quantities are unknown, its value can be calculated 
by the other three; as, 



be 

9X12 


d 

27 

b — _ 

ad 

4X27 


c 

12 

c — - 

ad 

4X27 , 


b 

9 

d = 

be 

9X12 


a 

4 


9, 


27 . 










IS 


Simple Interest. 


SIMPLE INTEREST. 

Interest is a profit on money which is lent for a certain time. 

Letters will denote. 

c — the standing capital, or lent money. 
r = interest on the capital c, 
p = per cent, on 100 in the certain time. 

Analogy, c:r — 100 : p. 

If p is the per cent, on 100, in one year, then t = time in years fbr the stand¬ 
ing capital c, and the interest r. 

Analogy, c : r = 100 : pt. 

From this analogy we obtain the equations, 


Interest, 

r — 

opt 

-. • • • 

100 * 

Per cent., 


100 r 

P — 

tc ’ 

Capital, 

c = - 

100 r 

- • - • 

pt 


- 1 , 
2 , 
. 3, 


Time in years. 


t 


100 r 
op 


Now for any question in Simple Interest, there is one equation which gives the 
answer. If the time is given in months, weeks, or days, multiply the 100 cor¬ 
respondingly by 12, 52,365. 

Example 1. What is the interest on $3789.35, for 3 years and five months, at 
6 per cent, per annum? 

i = 3X12+5 = 41 months, from the Equation 1, we have, 


Interest, 


r = 


3789.35X6X41 

12X100 


=776.81 Dollars. 


Example 2. A capital c = $469.78, gave an interest r = 150.72 dollars, hi a 
time t — 4 years and 7 months. Require the per centage per annum ? 

t = 4X12+7 = 55 months, from Equation 2, we have, 


Per cent.. 


12X100X150.72 
469.78 X 55 


= 7 per cent. 


Example 3. What capital is required to give an interest r = 345 Dollars in 6 
years, at 5 per cent, per annum ? From the Equation 3, we have, 


Capital, 




Example 4. A capital c = $2365 shall stand until the interest will be r = 550 
Dollars, at^> = 6 per cent, per annum. How long must the capital stand? 

From the Equation 4, we have, 


Time, 


t =- 


100X550 

2365^6 = 3 - 876 y 0ars - 


12X0.876 = 10.512 months, 4X0.512 2.048 weeks, the time t = 3 years, 10 

months, and 2 weeks. 



















Rebate or Discount—Fellowship. 


17 


REBATE OR DISCOUNT. 

Rebate or Discount is an allowance on money which is paid before due. 
a = amount of money to be paid in the time t. By agreement the amount is paid 
with a capital c, at the beginning of the time t, but discounted a Rebate r, at p 
per cent., so that the interest on the capital c, at p per cent., should be equal to 
the Rebate r, in the time t. a — c-\-r. 


Rebate, 
Capital, 
Per cent ,, 


Now, for any question in Rebate or Discount, there is one equation that will give 
the answer. 

Example 5. A sum of money, a = 78460 dollars, is to be paid after 3 years and 
6 months, but by agreement payment is to be made at the present time. What 
will be the Rebate, at 7 per cent. ? 

Rebate, r= 78460X7X3.5 = $45439 91> 

100+7X3.5 

FELLOWSHIP. 


r= apt 


c — 


100 -\-pt 
100 a 


100+p< 

100 (a — c) 
ct 


5. 

6 . 

7. 


Time, t= 1- Q °( a ~ c ). . . 8 . 

cp 

Amount, a — -^—(100 +»0. . 9. 

100 

Amount, a — — (100 + p t). . . 10. 

pt 


Fellowship or Partnership is a rule by which companies ascertain each 
fellow’s profit or loss by their stock. Each fellow’s part in the stock is called his 
share. The sum of shares is called the stock. 

Fellowships are of two kinds, Simple and Double. 

Simple Fellowship, when there is no regard to the time, the shares or 
stock is employed. 

Letters wiU denote. 


A = share of either one fellow. I S— stock or the sum of the shares. 

a = profit or loss on the share A. \ s = gain or loss on the stock S. 


Then, 

A:a = 

= S:s. 



Stock, S= 

a 

. 11. 

Share, 

. a S 

A = -. . 

s 

. 13. 

Gain or loss, s = 

A 

. 12. 

Profit or loss. 

a — A s 

S * 

. 14. 


Example 1. A person had invested A = $11645 in a stock S= $64800, which 
gave a gain of « = $13864. What will be the profit of the person’s share ? 


Profit, 


a = 11645 X 13864 =$2491.45. 
64800 


Double Fellowship. When the different shares are employed at a differ¬ 
ent length of time, each share is multiplied by its time employed, and the product 
is the effect of the share. 


Letters will denote. 


t = time for the employed share A. 

T— mean time for the employed stock S. 
e = effect of the share A. 


a = profit of the effect e. 
E— effect of the stock, 
s = gain of the effect E. 


Then, 


e : a — E: s. 


2 


















18 


Permutation. 


Formulas for Double Fellowship. 


Effect of A, 

„ a E 

s 

15. 

Time, 

t _ a E 

A s 

Profit of e, 

„ e s 

Cl — • —• • • • 

E 

16. 

Share, 

A — a ^ 
t s 

Effect of S, 

• 

• 

ii 

17. 

Meantime, 

rp _ C S 

a S' 

Gain of E, 

s=* aE 

o —*■ -• • • 

e 

18. 

Stock, 

S=^*. 
a T 


19. 

. 20 . 
21 . 

. 22 , 


Example 2. A canal is to be dug, and requires an effect E = 76850 (men and 
days) to be accomplished; after that it will give a gain s = 12390 dollars. An em¬ 
ployer has A =168 laborers. How many days must those laborers be employed at 
the canal, that the employer will obtain a profit a = 5000 dollars Z 


Time, t = 5000 X 76850_ =m6 day8t 

168 X12390 


PERMUTATION. 

Permutation is to arrange a number of things in every possible position. 
It is commonly used in games. 

Example 1. How many different values can be written by the three figures 
1, 2, 3. 

1 X 2 X 3 = 6 different values, namely, 

123,132, 213, 231, 312, 321. 

With any three different figures can be written six different values. Any three 
things can be placed in 6 different positions. 

Example 2. How many names can be written by the three syllables, mo, ta, la ? 
The answer is, Motala, Molata, Tamola, Talamo, Lamota, Latamo. 

Example 3. How many words can be written by the five syllables, mul, tip, li, 
ca, tion t 

1X2X3X4X5 = 120 words, the answer. 


COMBINATION. 


Combination is to arrange a less number of things out of a greater in every 
possible position. It is commonly used in games. 

Example 1. How many different numbers can be set up by the nine figures, 
1, 2, 3, 4, 5, 6, 7, 8, 9, and three figures in each number? 


9x8X7 

1X2X3 


84 different numbers. 


Example 2. How many different variations can a player obtain his cards, when 
the set contains 52 cards, of which he receives 8 at a time ? 

52X51 X50X49X4 8 X 4 7 X 16 X 45 = , 5253S150 wiations . 
1X2X3X4X5X6X-7X8 

If there are four players, and pr. 4 = 24, they can play 24X752538150 = 
18,060,915,600 different plays. 

If it takes half an hour for each play, and they play 8 hours per day, it will take 


18060915600 

2X3 


: 1128807225 days = 3092622 years. 












Alligation. 


19 


ALLIGATION. 

Alligation is to mix together a number of different things of different price 
or value, and ascertain the mean value of the mixture; or from a given mean 
value of a mixture ascertain the proportion and value of each ingredient. 

Let the different things be a, b, c and d, etc., their respective price or value per 
unit, z, y, x and w, etc. 

A = a + 6-f c-f (i, etc., the sum of the things. 

P— mean value or price per unit of A. 

Then, A P=az-\~by-\-cx-\-d w -f, etc. 

and P= az + by + cx + dw + :tc . 

A 


1 . 

2 . 


Example 1. If 3 gallons of wine, at $1.37 per gallon, 2, at $2.18, and 5, at $1.75, 
be mixed together, what is a gallon worth of the mixture ? 

A = 3-(-2-|-5 = 10 gallons. 

P= 3X1-37+2X 2. 18 . ± 6X 1.75 = $L72 per gallon . 

Alligation of two ingredients, a and b, with their respective prices or value per 
unit, z and y. z~^>P^>y. A = a-\-b. 

a : i> = (P — y): (z — P) .3. 

a = ~ V) and a = V) . . . 4 & 5 . 

(* = P) (*-y) 

Example 2. A silversmith will mix two sorts of silver, one at 54 and one at 64 
cents per ounce. How much must be taken of each sort to make the mixture 
worth 60 cents per ounce? (Formula 3.) P= 60. a; = 54. y = 64. 

a : b = (60 — 54) : (64 — 60) = 6 : 4, or 

4 ounces, at 54 cents, and 6 ounces, at 64 cents. 

Alligation of three ingredients, a, b and c, with their prices or value per unit, 
z, y and x. 

a ’: c' = (P— x) : (z — P) .. 6 

a" :b = (P—y) : (z — P) when 2 > P> y > x. . . 7. 

b:c" — (P — x)\{y — P) when z>y>P>x. . . 8 . 

a = a' + a", c = c' -f c • 

Example 3. A farmer will mix wheat, at 94 cents per bushel, with barley, at 72 
cents, rye, at 64 cents per bushel. How much of each sort must be taken to make 
the mixture worth 80 cents per bushel? 

(Formula 6 .) z = 94, y = 72, x = 64, and P= 80. 

a' : c , = (80—64) : (94 —80) = 16 : 14. 
a’ : b = (80—72) : (94 — 80) = 8 : 14. 

The wheat a — 16 -f 8 = 24 bushels, at 94 cents per bushel. 

“ barley 6 = 14 “ “ 72 “ 

“ rye c=14 “ “ 64 “ 

Alligation of four ingredients a, b, c and d, respective prices or value per unit, 
z, y, x and w. 


b ; c = (? - y]; lx _ pj } when * ; >: y -> p>x -> w > {iS: 

a , :d = (P— w): (z — P)) (11. 

a" : b = (P— y) : (z — P) Vwhen z~> P>^> x^>w •< 12. 

a'" :c = {P—x):(z — P)) (l3. 

a = a' + a" + a'". 

a:d r =(P— w): (z— P)) (14. 

b : d" = (P — w ) : (y — P) >when 2 r>?/>a;>P> w -i 15. 

c:d'" = (P—to): (* — Pj) (16. 

d =d f + d" + d f ". 

In the same manner, formulae can be set up for any number of ingredients. 








•20 


Arithmetical Progression. 


ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers, as 2, 4, 6, 8, 10, 12, 
&c., or 18, 15, 12, 9, 0, 3, in which every successive term is increased or dimin¬ 
ished by a constant number. 

Letters will denote, 
a = the first term of the series. 
b = any other term whose number from a is n. 
n — number of terms within a and b. 

S — the difference betw.een the terms. 

S == the sum of all the terms. 

In the series, 2, 5, 8, 11, a — 2, b = 11, n — 4, A = 3,and S = 26. 

4S?“When the series is decreasing, take the first term = b and the last term 
= a. 


The accompanying Table contains all the formulas or questions in Arithmeti¬ 
cal Progressions, and the nature of the question will tell which formula is to be 
used. 

Formulas for Arithmetical Progressions. 


a = b—S (n—1), • • 1, 

b—a 

* = * * 

- o, 

2 <S , „ 

a — — — b, • • - 2, 

» 

„ (b-f-a)(b— a), 

~ 2S —a— b , * 

.10, 

a = —’ —• • 3, 

n * 

* 2 ( S—an ) 

*“n(n-l)’ 

• 11, 

b = a+A (» — 1), * -4, 

, 2 {bn-S) > 

n (n—1) 

- 12, 

. 2S , 

0 1 - " dy • • • Oj 

n 

s = n(a±b\ . . 

2 

- 13, 

. b = -—1)> * * 6, 

n 2 

(a-fb)(b+A—a), 

2A 

.14, 

» = 6 -=?+l, ... 7, 

S—n £a+- (»—1) J 

15, 

t 2 S . 

a+b’ 

^ ==n [ 6 _i (n _l ) ] . 

- 16, 




“=4±\/( 

b+l) 2 -2SS, * * * * 

17 , 


&— — ^ -\-2SS, - * • 

- 18 , 

* l - 1 a In «N 2 , 2,sr 

2 (g -+ T» * 

19, 

1 

_ 1 , b , //I fc\ 2 2 S 

"”2 - + i±\/(- 2 +r)-—. • • 

• 20 . 


























Arithmetical Progression. 


21 


Example 1, A man was engaged to dig a well at one dollar ($1) Sir the first 
loot of the depth of the well, $1*84 for the second, and 84 cents more per every 
successive foot in depth, until he reached the water, which was found at a depth 
of 25 feet. How much money is due to the man ? 

This will be answered by the formula 15, in which a — 1, d = 0'84, and 
n — 25, then the sum, 


= 25 [ 1+ ^|- 4 (25-l)] 


$277 the answer. 


Example 2. A Propeller ship which is to run between Philadelphia and 
Charleston, cost $116500, of which the company agreed to pay on account 
$1407 5 at her first trip to Charleston; and per every successive trip, they paid 
$650 less than the former. How many trips must the vessel make until she is 
fully paid ? 

This will be answered by the formula 20, in which b = $14075, d = 650, and 
S = 116500. 



1 i^L 5 _. A/14075 1 \ 

650 'y/ V. 650 + 2 ' 


* 2XH6500 

650 


10-6 or 11 trips. 


Arithmetical Progressions of a Higher Order. 

Arithmetical Progressions are of the first order, when the difference S is a 
constant number, but when the difference S progresses itself with a constant 
number, the Progression is of the second order. 

When the difference $ progresses in a second order, the Progression is of the 
third order, &e., Ac., and is thus explained: 

1, 2, 3, 4, 5, 6, . . . n, • • Arith. Prog., first order. 

1, 3, 6, 10, 15, 21, . . . n J n+ l 2 .... 2d. order. 

2 


1, 4, 10, 20, 35, 56, 


1, 5, 15, 35, 70, 126, 


n (n+l)(n+2) 
2X3 ’ 

n (tt+l)(n+2)(TO+3 ) 

2X3X4 * 


■> 3d. order. 


4th. order. 


Here you will discover that the sum of n terms in one order, is equal to the 
same nth term in the next higher order. Arithmetical Progressions of the first, 
second, and third orders, are applied to 

PILES OP BALLS AND SHELLS. 

Triangular Piling. 

Example 1. A complete triangular pile of balls has n — 12 balls in each side. 
Require how many balls in the base, and how many in the whole pile ? 

12 f! 2+1) 

~ 2 


In the base, 

Whole pile, • 

1, 4, 9,16, 25, 36, - 
1, 5, 14, 30, 55,91, • 


12 (12+l)(12+2) 

2X3 


= 78 balls, 

364 balls, - 


2 d. order. 


Square Piling. 
»2 - 

n(n+lX2ft+l) 

2X3 9 


3d. order. 

2d. order. 
3d. order. 


[See Examples 2 and 3 on page 23 J 


















22 


Geometrical Progression. 


GEOMETRICAL PROGRESSION. 

Geometrical Progression is a series of numbers, as 2:4 :8 :16:32 : Ac., 
or 729.243: 81: 27 :9 : &c., in which every successive term is multiplied oi divided 
by a constant factor. 

Letters will denote, 

a = the first term of the series. 
b — any other term whose number from a is n. 
n = number of terms within a and b. 

r — ratio, or the factor by which the terms are multiplied or divided. 

S = Sum of the terms. 

In the series 1:3:9: 27 : a = 1, b = 27, n — 4, r = 3, S = 40. 

The accompanying Table contains all the formulas or questions in Geometrical 
Progressions. The nature of the question will tell which formula is to be 

used. 

Formulas for Geometrical Progressions . 


b 

a - --, • 

—1 

m m 

1, 

n ~ 1 nr 

r = \ / —, 

\/ a 

• 7, 

a = S — r(S~ 

-6),. - 

2, 

S—a 

r S — b* * * * 

8, 

cv r -1 

• ® 

3, 

ar»-f S — rS — a = 0, 

• *, 

b = ar *— l , 

» • • 

4 , 

S r — 1» * * * 

10, 

b-s s ~ a 


K 

1 

It 

1 

M 

'w' 

> 

1 

•11, 

r 

9 * * 


r — 1 1 

0 = 

(rn-1, 

6, 

5 (r» — 1} 

(r — l)r”— 1 * 

12, 


n _ 1 log.b — log. a 
log.r ’ 


• • 


n *= 1+ 


log.b — log.a 


log.(S+a) — log.(S — by 

log.{a+S(r —1 )] — log.a 
= log.r. * 

_ 1 log.b—log.\br — S(r— 1)] 
log.r. * 


n—1 


8 => 




*— 1 


\fb '— n ~\ r a 


18 , 

14, 

16 , 

10 , 

V, 



















Geometrical Progression. 


2a 


Example 1. Required tlie 10th term in the Geometrical Progression 4:12:3t,.„ ? 
Given a = 4, n — 10, and r — 3. We have, 

Formula 4. b — ar™— 1 = 4X3® = 78732, the tenth term. 

Example 2. Required the sum of the 10 terms in the preceding example l 

Formula 11, £ = —— = 118096, the sum. 

r — 1 2 

Example 3. Insert 6 proportional terms between 3 and 384 ? 

Given a = 3, b — 384, and n — 6+2 = 8. 

n—l 

Formula 7, r — 




then 


384 „ 

T = 2 ’ 


3 : 6 :12 : 24 : 48 : 96 :192 : 384, the answer. 


Example 4. A man had 16 twenty dollar gold pieces, which he agreed to ex¬ 
change for copper in such a way, that he gets one cent on the first $20, two on 
the second, four on the third, and eight on the fourth, &c., &c.; until the sixteen 
$20 pieces were covered. How many cents will come on the sixteenth gold 
piece, and what will be the whole amount of copper on the gold? 

In the progression 1: 2 : 4: 8 : &c., we have, 

Given n = 16, r =2, and a = 1, then, 


„ , 21 ° 48 16 « 266 ® 

b = 1X2 16 1 = — = —— —- = = 32768 cents, on the 

& 2 2 Z 


Formula 4. 
sixteenth piece. 

The total sum of cents will he found by the 


Formula 10. 


S= 


32768 X2—1 

2—1 


= 65535 cents = $655-35. 


Piling of Balls and Shells.—[From page 21.] 

Example 2. How many balls are contained in a complete square pile, n = 10 


rows? 


10(10 +1)(2X10+1) .10X11X21 


2X3 


= 385 balls. 


Rectangular Piling. 


Let m be the number of balls on the top of the complete pile, and n = num¬ 
ber of rows in the same, then the number of balls in the whole pile will 

be, 


n(n+l)(2n+3m—2) 

2X3 * 


3d. order. 


The number of balls in the longest bottom side will be = wi+n — L 

Example 3. The rectangular pile having 15 rows and 23 balls on the the top, 
how many in the whole pile? 

1 5 (15+1)(2XI5+3X23-2) 15X16X67 = 26g0 bal]g 

2X3 6 
















24 


Compound Interest. 


COMPOUND INTEREST. 

Compound Interest is when the interest is added to the capital for each 
year, and the sum is the capital for the following year. 


Amount, a = c(l -f p) n . 


Capital, 


c — 


1 . 


2 . 


Percentage, 


y-V 7 ”-!. .3. 


Number of years, n = — \23fL, 

log.(l +p) 


4. 


(1 +P) n 

4®=* In these formulas p must be expressed in a fraction of 100. 

Example 1. A capital c = 8650 standing with compound interest, at p = 5 per 
cent. What will it amount to in n = 9 years ? 

Amount a — 8650 (1.05.) 9 = 18419 dollars. 

Example. 2. A man commenced business with c = 300 dollars; after n — 5 years 
he had a = 6875 dollars. At what rate did his money increase, and how soon will 
he have a fortune of 50000 dollars? 

The first question, or the percentage, will be answered by the formula 3. 

5/6875 _ 1== f/ 22.9166 — 1 = 0.87, or 87 per cent. 

K 300 v 

The time from the commencement of business until the fortune is completed 
will be answered from the formula 4. 


n-- 


log .50000 — log.300 4.69897 — 2.47712 


log. 1.87 
or 8 years and 2 months. 


0.2720048 


8.169 years, 


Compound Interest Table, calculated from formula 1. 


M 

Compound Interest. 

Tears. 

5 per ct. 

6 per ct. 

7 per ct. 

1 

1.0500 

1.0600 

1.0700 

2 

1.1025 

11236 

1.1449 

3 

1.1576 

1.1910 

1.2250 

4 

1.2155 

1.2625 

1.3108 

5 

1.2770 

1.3382 

1.4025 

6 

1.3400 

1.4185 

1.5007 

7 

1.4071 

1.5036 

1.6058 

8 

1.4774 

1.5938 

1.7182 

9 

1.5513 

1.6895 

1.8385 

10 

1.6289 

1.7908 

1.9671 

11 

1.7103 

1.8983 

2.1048 

12 

1.7958 

2.0122 

2.2522 

13 

1.8856 

2.1329 

2.4098 

14 

1.9799 

2.2609 

2.5785 

15 

2.0789 

2.3965 

2.7599 

16 

2.1829 

2.5403 

2.9522 

17 

2.2920 

2.6928 

3.1588 

18 

2.4066 

2.8543 

3.3799 

19 

2.5269 

3.0256 

3.6165 

20 

2.6533 

3.2071 

3.8697 

21 

2.7859 

3.3995 

4.1406 

22 

2.9252 

3.6035 

4.4304 

23 

3.0715 

3.8197 

4.7405 

24 

3.2251 

4.0487 

5.0724 

25 

3.3864 

4.2919 

5.4274 

30 

4.3219 

5.7435 

7.6123 

35 

5.5166 

7.6861 

10.6766 

40 

7.0400 

10.2858 

14.9745 

45 

8.9850 

13.7646 

21.0025 

50 

11.6792 

18.4190 

29 4570 

60 

18.6792 

32.9878 

57.9466 


This table shows the value of one unit of 
money at the rates of 5, 6 and 7 per cent, per 
annum, compound interest, up to 60 years. 

Example 1. What is the amount of 864 pounds 
sterling for 12 years, at 6 per cent, compound 
interest? 

Table, 2.01219 x 864 = 1738.53216, or £1738 
10s. 7.7 d. 

Example 2. What is the amount of 3450 dollars 
for 18 years. at 5 per cent, compound interest? 

Table, 2.40661 X 3450 = 8302.80 dollars. 

When the interest is compounded in more or 
less than one year, at the rate of interest per 
year, and m = the number of months in which 
the interest is compounded; 

Then, instead of p in the formulas, put m P , 

12t? 12 5 

and instead of n, put -—-* 
m 

Example 3. A capital of 500 dollars bears 
compound interest semi-annually, at 5 per cent, 
per annum; what will it amount to in 10 years? 


0.05 


= 0.025 


m = 6 months, p — ^-2* = 

12 12 

and n = 12 > > - 1Q = 20, 

6 

then, a = c(l + p) n = 500(1 + 0.025)20 = 
8193.11 dollars, the answer. 

log.a + 0.025) = 0.0107239 

20 


Amount , 


log. 500 = 
8193.11 = 


0.2144780 

2.6989700 

2.9134480 



















Annuities. 


25 


ANNUITIES. 

Annuity is a certain sum of money to be paid at regular intervals. 

A yearly payment or annuity 6 is standing for n years; to find the whole amount 
a, at p per cent, interest. 


. 1 . 


Amount, a = 6nj~l-f Simple Int. 

Amount, a = ~ (1 + p) n — 1J Comp. Int.2. 

A yearly payment or annuity & is to be paid for n years; to find the present 
worth, or the amount a, which would pay it in full at the beginning of the time 
n, deducting p per cent, interest. 

Amount, 


Amount, 


a = bn £l — ^J Simple Int. 
a = b -[l 

p L 


--ICornp. Int. 


(1 + P) n J 


3. 

4. 


A debt D, standing for interest, is diminished yearly by a sum b; to find the 
debt d after n years, and the time n when it is fully paid ? 

The debt d after n years will be— 


d= lRp=M L±^LL& Comp. Int. 
P 


The time n until fully paid will be— 


n= log-b — log.(b — D p) 
log.( l +p) 


5. 


6 . 


If b= D p, then n— co, or the debt D will never be paid. If &<Z> p, the debt 
D will be increased. 

To find the yearly annuity 6 which will pay a debt D in n years, at p per cent, 
compound interest ? 

DP^+pf 1 7 

(l+pf-l 

Annuity Table, 

Showing the present worth of an annuity or rent of one unit of money, at 5, 6 
and 7 per cent, compound interest for years up to 60, calculated from formula 4. 


Tears. 

6 per ct. 

6 per ct. 

7 per ct. 

Years. 

5 per ct. 

6 per ct. 

7 per ct. 

1 

0.9524 

0.9434 

0.9345 

17 

11.2741 

10.4772 

9.7632 

2 

1.8594 

1.8333 

1.8080 

18 

11.6896 

10.8276 

10.0591 

3 

2.7232 

2.6730 

2.6243 

19 

12.0853 

11.1581 

10.3356 

4 

3.5459 

3.4651 

3.3872 

20 

12.4622 

11.4699 

10.5940 

5 

4.3295 

4.2123 

4.1001 

21 

12.8211 

11.7641 

10.8355 

6 

5.0757 

4.9173 

4.7665 

22 

13.1630 

12.0416 

11.0612 

7 

5.7864 

5.5824 

5.3892 

23 

13.4881 

12.3034 

11.2722 

8 

6.4632 

6.2098 

5.9712 

24 

13.7986 

12.5503 

11.4693 

9 

7.1078 

6.8017 

6.5152 

25 

14.0939 

12.7833 

11.6536 

10 

7.7217 

7.3601 

7.0235 

30 

15.3724 

13.7648 

12,4090 

11 

8.3064 

7.8868 

7.4986 

35 

16.3742 

14.4982 

12.9476 

12 

8.8632 

8.3838 

7.9426 

. 40 

17.1591 

15.0463 

13,3317 

13 

9.3936 

8.8527 

8.3576 

45 

17.7741 

15.4658 

13.6055 

14 

9.8986 

9.2950 

8.7454 

50 

18.2559 

15.7618 

13.8007 

15 

10.3796 

9.7122 

9.1079 

55 

18.6334 

15.9905 

13.9399 

16 

10.8378 

10.1059 

9.4466 

60 

18,9292 

16.1614 

14.0389 





















26 


Calculus of Differentials' 


By the Differential Calculus we ascertain the simultaneous progress 
of variable quantities depending on one another. The variable quanti¬ 
ties are designated by the last letters u, v, x, y, z, and the constant 
quantities by the first, a, b, c, e, /, of the alphabet. The letter d is placed 
before variables to denote the instantaneous progress of that quantity, 
as dx, and called the differential of x. d • reads differential. Let the side 
of a square be denoted by x and the area by z ; when x increases uni¬ 
formly, 2 will increase more rapidly. When x = 1, z =1, but when x = 2, 
2 = 4. Whefi we know the instantaneous increase of x, what will be 
that of zl If we add, say only a point to the side x, there will be added 
two lines or 2 a? to the square. We know that z = # 3 , the d‘ or increment 
of the square will be dz — 2xdx, of which dx is the point added to x and 
2 a? is the two lines added to the square, called the differential coefficient. 
Let v denote the volume of a cube, and x its side, we haye v =x 3 and 
dv = ^x^dx, which shows that if a point dx is added to x there will be 3a; 3 
or three squares added to the cube. 

The d ■ of any power of a variable is equal to the power diminished by 
1, multiplied by the primitive exponent and the product by the d‘ of the 
variable. The d' of a constant is = o. When the constant is a factor to 
the variable it appear unchanged in the d • coefficient, but when a term 
it disappears. 

I. The d • of length u of any line defined by a formula of rectangular 
co-ordinates x and y , is du = ^ /dx--\-dy^. II. The d • of area z of any 
plan figure bounded by a curved line and rectangular co-ordinates is 
dz=y dx, y = ordinate, x = abscissa. III. The d • of solidity v of any 
figure bounded by a plan rotating round its abscissa a ?, is dv = it ?/ 3 dx, 
y — ordinate of the outer line of the plan. IV. The d- of surface z of 
any solid bounded by a plan rotating round its abscissa x, is dz = 2 n y du, 
in which u = length of the outer edge of the plan. 

Successive d’S is when the first d' coeff. is considered a function of a 
new function. 

du l du \ d?u 

u = ax*. 1st. d • coeff. — =4 ax 3 , 2nd. d- coeff. d — )=—=12 ax' 1 , 

dx \dx] dx 3 

_ __ , / d^u \ d 3 u 

3d. d • coeff. d I ) = — 3 = 2 4 a x, etc., etc. 


d?u means the second, d?u the third d’ coefficient of u. dx 3 means the 
square of the d • of x, etc. 

Example 1. The diameter of a sphere increases at a rate of dx= 2-31 
inches per second, when x== 9 5 inches, at what rate (dv =?) does the 

TT 00^ 

volume v increase! v= —— = 0-523a; 3 , dv = 0-623X3 x"-dx = 1*669X2'31 

6 


= 327-1 cubic inches, the answer. 

Example 2. It is found that the displacement of a ship increases as 
a 1 -® the draft of water. At the load draft a= 18 feet the displacement is 
T=2000 tons. Required the displacement (t = l) when x= 12 ft. and how 
much (dt. 1) can the vessel be loaded pef dx = 1 inch or 1-12 foot, at that 
draft, 

»a.5 r p 12 15 V2000 

t =- — s= - - — = 1088-6 tons, at x = 12 feet draft. 

a 1,1 18'-* 


dt = 


1-5 Tx%dx 
a l,i 


1-6X2000XJ/12 
18‘- 6 X12 in. 


11-34 tons per inch. 


The following page contains the differentials of formulas and trigo¬ 
nometrical functions. I means the Naperian logarithm. The common 
logarithm log. multiplied by 2-302585 gives the Naperian logarithm l. 

















Calculus Differential Formulas. 27 




































28 


Calculus of Integrals. 


The Integral Calculus is the reverse of the Differential, or to find 
the original formula of a given differential. The symbol f is placed be¬ 
fore tue d' to denote that the integral is to be taken out of it, or that the 
original formula is to be found. 

3 a 

The d 1 of a x 3 =3 a dx, and fZax^dx — --— —ax 3 . 

3 

Rule to find the integral. Add 1 to the exponent of the variable x in 
the d-, divide the d - by the new exponent, dx will disappear, and the 
quotient is the integral. The integral / does not effect a constant. A 
constant term in a formula disappears in its db, consequently any inte¬ 
gral may have a constant term, whose value is determined by making 
the variable in the integral = o, when the first member in the formula 
will be a constant. It is therefore customary to add a constant C to the 
integral. When it is known that the first member is = o at the same 
time the variable in the/is — o, then C= o. When a differential is to 
be integrated between two limits of the invariable, say x — a and x = b, 
it is indicated by / b . or / b 3 c x' 2 dx = c(b 3 — a B ). 

a a 

Successive differentials are accompanied with the same order of inte¬ 
grals, as ff 6 x dx 3 = /3a >2 dx — x 3 . 

The integrals of the differentials gives the formula for the problem. 

Example 3. It is required to find by the calculup a formula for the 
area «ofa rightangled triangle. Proposition II, page 78. dz = y dx, the 

formula for the hypothenuse is y =ax, dz—axdx, z=fa x dx = 
or the area z is half the rectangle of the sides x and y. 

Example 4. Find a formula for the convex surface z of a cone, whose 
side is u, and r — radius of the base? Prop. IV, page 78. dz = 2nrdu, 
and z = J’2nrdu = nru, the answer. 

Example 5. Find a formula for the area z of a circle, when it is known 
that the circumference?/= 277® I Prop. II, page 78. dz = y dx = 2n x dx, 
and z=f2nxdx = nx' i the answer, x = radius of the circle. 

Example 6. Find a formula for the area sofa parabola of x = abscissa 
or height, and y = ordinate, or half the base? Formula for a parabola 


y = fp x, in which p = the constant parametric diameter, or p = 


y j 

® = —, dx= 
V 

2 x y 


2 y dy 


— ■ Prop. II, dz = y dx= 


2 y 1 dy, 


and z 


=/ 


r 

X. 


2 y 3 dy 2 y s 


p * p d p 3 p 

, the answer, or the area of a parabola is % of the base by the height. 

Example 7. Find a formula for the volume v of a paraboloid] Prop. 

HI. V* & = —^, and v=f 3 ” ^ = _IA but p = ^and „ = 

p dp 2 p * X 

^-y 1 x, the answer. 

V. The center of gravity * from the origin of x, of any plan figure 
bounded by a curved line and rectangular co-ordinates is s = -—^—’ 
when z area of the plan. 

VI. The center of gravity 8 from the origin of x of any solid figure is 

fee z dec 

s=z ——-> when z = ordinate cross section and v volume of the same. 

v 

Example 8. Find a formula for the eentre of gravity (s = I) of a cone. 
The ordinate cross-section z — Tcy 1 , and v= x, when x = height and 

O 

y radius of the base of the cone. Prop. VI. s = - X - ~ ^ 

V TV f 1 x 

As the center of gravity is not influenced by the proportion of x and y, 

3 f tt txP dec 3 77* ec^ 

we can make y = x, when s = ——-- x , from the top. 

- 3 4 77 X 3 ’ 1 


77 X ° 






















29 


Calculus Integral Formulas. 


DIFFERENTIALS. 

INTEGRALS. 

DIFFERENTIALS, 

. INTEGRALS. 

fdX—X-\-x) 

fxdx= ~ f-C, 1 

p dx 

J v/^T® a 

l‘{x-\-ya^x 1 ), 21 

i 

flax 3 dx=4afx i dx = ax*- j-C, 2 

b 

jZmx^dx ** 

m b 3 — m a 3 , 22 

fx n dx , = 

a;»+i 

»+i+ c - 3 

6 

/m a; da; — 

j(b*~ o 2 ), 23 

J\/xdxfx^dx = 

3 +C * 4 

00 

/* dx 

h 23 


2^i+C, 5 

/a da; 

v * 

/ s/a'—x* 


Z*®-f- C, 6 

b a 

{—< 

/=/+/ 26 
a a b 

/*da; /»-3. 

J *r=J* <*» = 

“ ^ + C ’ 7 

/sin. a: da- = - 

- cos.a+C, 27 

y^-/^= 

-i+ c ' 8 

/cos. a:da: = 

sin.a;-f- C, 28 

f( ax 3 +—)dx 

J ' 2J/0? 7 

° ^ i 6i/S4-C 9 

/tan. a: dx = ■ 

— Z- cos.a+C, 29 


/a da: 

J X 

al’x-y C t 10 

/ cot. a; da; = 

— Z- sin.a+C, 30 

Pbdx 

J a-j-x 

6Z-(o+a;)-f-C, 11 

/»da; 

/sin.a; 

Z* tan. + C, 31 
z 

/»3 a a; 3 da; 

J b-\-ax i ~ 

l"(b-\-a a: 3 )-j-C, 12 

f—— = Z* tan. (-7 + ^-)+£,32 
«/ cos.a; \ 4 1 2 / 1 ’ 

faxdx-\-'ix 1 dx-b’ 1 dx 


/sin .x cos .xdx = 

— sin 3 . a+C, 33 

/(a 3 -f-& 9 ) da: = 

a;(a 3 +6 3 H-C, 14 

n&m.bx , 

y—* —= 

T - 

//T^ Xr3 . 

f(ax—2x^fdx = a: 3 (-—aa;-}- —J-f-C, 

00 t 
/’cos.fia; 

/ -d» = 

%/ Su 

GO, 35 

/3(a a;—a; 3 ) 3 (a— 2x)dx =(ax—x 2 ) 3 -\-C, 

= circle arc of which Z=tant. 

Pn^x^dx) 
y/ a-\-x n 


/* — Airnlp aro. nf wTiir>h 

y d 1 -\-x n -\-Cy 17 

% 

1 

a; = sin. versus. 37 

r2adx 

Ja?-^x* ~ 

l.^+C, 18 

a—x 

fff6adx 3 =ff6axdx^=fiax' i dx=ax 3 -f C 

/ yd^fx 1 dx == — (/a 5 -f-a; 3 -f Z-(a: -f- 

//2(o+&)da; 3 = 

(a-yb)x*-\Oy 39 

/y'a^Piarda; = — 

3 

b(ya-ybx) s -\-Cy 20 

JJ2v 3 do; 3 + Sv xdx dv -\-2 x^di^—x v s ,40 















Maxima and Minima. 


jO 


Two variable quantities a; and y depended on one another, to find the 
value of one, when the other is a maxima or minima. 

! t dx 
is a maxima or minima when its J dy 
first differential coefficient- 1 dy 

u=°* 

When the second d • coef. is positive, y is a minimum, and when 

negative y is a maximum. The variables may have both maximums and 
minimums, as formulas will indicate. 

Example 1. Find the value of x when y is a maximum or minimum, in 
the formula y = x 3 —12 x -[-22'? dy = ( 3# 2 —12 ) dx, 

dOC 


= 3 a?— 12 = o. 


Of which x = v/ J«r 2 -=2 the answer. —= 6x, which is positive, conse- 


d?y 

-7T~ = * WiC 0,11 »W Cl . — 7T 

J dx 2 

quently y — 2 3 —12 X 2 -f- 22 = 6, a minimum, when x = 2. 

Example 2. It is required to cut out the strongest possible beam of 
height h and breadth b, from a log of diameter D, fig. 221 page 1741 The 
strength of a beam is in proportion to bh? which is to be a maximum. 

U^W + h?, h^D^-P, bK i =b{D^—h’), d(bh !1 ) = (D n -—3b' i ) db. 

d (bh?) _ 

- , — = D " 1 — 3 b 2 =o, of which the breadth b—Dy / % = 0-577 D, and height 

d 0 

D-y/0 6666 =0-8164 D, the answer. The second d' coef. 

d?(bh?) 

- ^ — = — 6 b, which is negative, and therefore bh? is a maximum 
when b = 0-577 D. 

Example 3. It is required to know the proportion of heighth h and 
diameter D of a cylinder, having the greatest cubic containt v, with the 

TT D 2 2 V 

smallest surface 'z including top and bottom 1 »= — —+ nDh, 

Z il 

2 

which is to be a minimum. Set v = l and D= 1, then z=—-\-irh, and 


dz = (ir — ^jdh, 


dz 
dh " 


h? 


— o, when 




1284 D, the answer. 


The second d • coef. ^ = 4- which is positive, and z a minimum when 

dh? ‘ h* 

h = 1-1284 D. 

Maclaurin’s Theorem. 

Maclaurin's Theorem , explains how to develop into a series a function 
with one variable, as 

, „ , x /du\ x l /d-u\ , x 3 / d 3 u\ , x * / d"U\ , 

U + T \dx ) + ~2\d&) "^2X3W^ ”^2X3X - n'dw) 6 

where the factors in the parenthesis is that which it assumes when x=o. 

The function u — —j— developed into a series will be 

CL "p QD 

1 1 X , X* X 3 x n 

— =-; + -7——7-,. etc. 

a-\- x a a? a? a 4 a*+ l 

Taylor’s Theorem. 

Taylor's Theorem, explains how to develop into a series a function of 

the sum or difference of two variable as u = 

_L . \du , d?u y* \d 3 u y 3 , d u 
F(a; :y)=M±-y+ d —. *2 yTSyn’ 

where u represents the value of the function when y — o. 















Interpolation. 


81 


Interpolation is to insert numerical values between given data, for 
constructing tables or empirical formulas expressing the probable rela¬ 
tive variation of quantities. Let * and y be two variable quantities de¬ 
pending on one another and measured in simultaneous stages of their 
progress, as 

x x x. 2 x 3 x 4 and x & 

Vi 2 h 2/3 2/4 and 2/5 


We have y — Ay-cYBy^Cyz-VDy^Ey-Y&Q,. - - - - 1 

2 3 4 6 given 

V V V Vdata. 

^_ (g — (as — x 3 ) (x — x 4 ) (x — x 6 ) ' 

(xx—x. 2 ) [x x —x 3 ) (xx—x 4 ) (xi— x,) \ 


— {x—X 3 ) (x — x 4 ) [x — x a ) | 

~{x. 2 —X 1 ) [x 2 —x 3 ) (x,— x 4 ) (x>— x 5 ) , 

3 2 > - - --1 I I • 

'O _(*— ^ — X ^ ’ 

§ _ (x 3 — Xx) {x 3 — X 2 ) (x 3 —x 4 ) ^(x 3 —x 5 ) | 

^ __ (as— xi) (ag — x 2 ) {x — xj {x — x b ) \ 

Z ~{x 4 — X 4 ) (x 4 — x. 2 ) {x 4 —x 3 ) {x 4 — x 6 ) • 

<D i 

-Q 4> - -- -- -- -- -- — 1 

I (x — Xx) (x—x 2 ) (x — x 3 ) (x—x 4 ) • 

(Xtj Xx ) (x 3 X 2 ^ {x 3 X 3 ^j [Xr, x^ 

5>.. J 


The values of the coefficients A, B, C, P, and E, with their given data, 
inserted in formula 1 gives an empirical formula for the variation of x 
and y. The number of observations or given data of x and y should be 
one more than the order of progression. In arithmetical progression 
two observations are sufficient for a correct formula. For all curves in 
the conic sections, or others which are of the second order, there should 
be at least three observations. Pressure of steam progresses with the 
temperature in the 6th order, for which requires seven observations to 
make a correct formula. When the order of progression is not known, 
the more observations gives the most correct result. 

Example. Let y represent the boiling-point of salt water and * the 
percentage of salt in solution. It is found in three experiments, 


that x\= 3, *2=18, *3=36 per cent. salt, 

when 2/1=213-2 ^2=219°, 1/3=226° boiling-point. 

Find a formula that will give any intermediate value of * and y 1 


—18)(*—36) (*—3)(*—36) ^ (*—3)(*—18) 

(3—18) (3—36)’ ~ (18—3)(18— 36)’ (36—3)(36—18)’ 

y=213-2 44-219 P4-226 C. y= 0-4*-}-212 





















32 


United States’ Standard Measures and Weights 


-- ! - 1 

UNITED STATES’ STANDARD MEASURES AND WEIGHTS. 

MEASURE OF LENGTH. 

The Standard Measure of"Length is a brass rod = 1 yard at the temperature of 
32° Fahrenheit. The length of a pendulum vibrating seconds in vacuo, at 
Philadelphia is 1-08614 yards, at -f 32° Fahrenheit. 

The Surveying Cliain is — 22 yards = 66 feet. It consists of 100 links, 
and each link = 7-92 inches. 

ROPES AND CABLES. 

1 Cable length = 120 fathoms = 720 feet. 

1 fathom = 6 feet. 

GEOGRAPHICAL AND NAUTICAL MEASURES. 

1 Degree of the great circle of the Earth round the Equator = 69*032 statute 
miles = 60 Nautical miles. 

1 Statute mile = 5280 feet = 0*86875 Nautical miles. 

1 Nautical mile = 6037*424 = 1*150 Statute miles. 

LOG LINE. 

The Log Line should he about 150 fathoms long, and 10 fathoms from 
the Log to the first knot on the line. If half a minute glass is used, it will be 
51 feet between each succeeding knot. For 28 seconds glass it will be 47*6 feet 
= 7-93 fathoms per knot. This is the length of knot by calculation, but prac¬ 
tically it is shortened to 7'5 fathoms per knot for 28 seconds glass. 

MEASURE OF CAPACITY. 

Gallon* The standard Gallon measures 231 cubic inches, and contains 
8-338S822 pounds Avoirdupois = 58372-1757 grains Troy, of distilled water, at its 
maximum density 39-83° Fahrenheit, and 30 inches barometer height. 

Bushel. The standard Bushel measures 2150-42 cubic inches = 77*627413 
pounds Avoirdupois of distilled water at 39-83° Fahrenheit, barometer 30 inches. 
Its dimensions are 18£ inches inside diameter, 19£ inches outside, and 8 inches 
deep; and when heaped, the cone must not be less than 6 inches high,equal 
2747-70 cubic inches for a true cone. 

Pound* The standard Pound Avoirdupois is the weight of 27*7015 cubic 
inches of distilled water, at 39*83° Fahrenheit, barometer 30 inches, and 
weighed in the air. 


MEASURE OF LENGTH. 


Miles. 

Furlongs. 

Chains. 

Rods. 

Yards. 

Feet. 

Inches. 

1 

8 

80 

320 

1760 

5280 

63360 

0125 

1 

10 

40 

220 

660 

7920 

0-0125 

01 

1 

4 

22 

66 

792 

0 003125 

0025 

0-25 

1 

5.5 

16-5 

198 

000056818 

0 0045454 

0*045454 

0*181818 

1 

3 

36 

0 00018939 

000151515 

001515151 

0-0606060 

0-33333 

1 

12 

0000015783100001262620-001262626 

0-00505050 

0-0277777 

0083333 

1 


MEASURE OF SURFACE. 


Sq. Miles. 

Acres 

S.Chains. 

Sq. Rods. 

Sq. Yards. 

Sq. Feet. 

Sq. Inches. 

1 

640 

6400 

102400 

3097600 

27878400 

4014489600 

0001562 

1 

10 

160 

4840 

43560 

6272640 

00001562 

01 

1 

16 

484 

4356 

627264 

0-000009764 

000625 

0 0625 

1 

30-25 

272-25 

39204 

0000000323 

00002066 

000-2066 

00330 

1 

9 

1296 

00000000358 

000002296 

0-0002296 

0-00367 

0-1111111 

1 

144 

0-00000000025 

0 000009169 000000159(0*00002552 200007716 

0 006944 

1 






























Measure of Capacity and Weights. 


33 


MEASURE OF CAPACITY. 


Cub. Yard. 

Bushel. 

Cub. Feet. 

Pecks. 

Gallons. 

Cub. inch. 

1 

21 6962 

27 

100-987 

201-974 

46656 

0-03961 

1 

1-24445 

4 

9-30918 

2150-42 

0-037037 

0-803564 

1 

3-21425 

7-4805 

1728 

0-009259 

025 

0-31114 

1 

2-32729 

537 605 


0-107421 

0133681 

0-429684 

1 

231 



0-000547 

0-001860 

0-004329 

1 


MEASURE OF LIQUIDS. 


Gallon. 

Quarts. 

Pints. 

Gills. 

Cub.inch. 

1 

4 

8 

32 

231 

0-25 

1 

2 

8 

57-75 

0-125 

0-5 

1 

4 

28-875 

0.03125 

0125 

0-25 ' 

1 

7-21875 

0004329 

0-017315 

0-03463 

0-13858 

1 


MEASURES OF WEIGHTS. 

AVOIRDUPOIS. 


Ton. 

Cwt. 

Pounds. 

Ounces. 

Drams. 

1 

20 

2240 

35840 

573440 

0-05 

1 

112 

1792 

28672 

0-00044642 

0-0089285 

1 

16 

256 

0-00002790 

0-000558 

00625 

1 

16 

0-00000174 

0-0000348 

0-0016 

0-0625 

1 

TROY. 

Pounds. 

Ounces. 

Dwt. 

Grains. 

Pound Avoir. 

1 

12 

240 

5760 

0-822861 

0-083333 

1 

20 

480 

0-068571 

0-004166 

0-05000 

1 

24 

0-0034285 

0-0001736 

0-002083333 

0-0416666 

1 

0-00014285 

1-215275 

14-58333 

291-6666 

7000 

1 

APOTHECARIES. 

Pounds. 

Ounces. 

Drams. 

Scruples. 

Grains. 

' 1 

12 

96 

288 

5760 

008333 

1 

8 

24 

480 

0-01041666 

0*125 

1 

3 

60 

0-0034722 

0-0416666 

0-3333 

1 

20 

0-00017361 

0 0020833 

0-016666 

005 

1 














































S4 


Money. 


MONEY AND COINS OF THE UNITED STATES. 


10 mills = 1 cent. 
10 cents = 1 dime. 


10 dimes 
10 dollars: 


: 1 dollar. 

1 eagle. 


The standard gold and silver coins contain 900 parts of pure metal and 100 parts 
of base metal in 1000 parts of the alloy. 

The remedy of the Mint is the allowance for deviation from the exact standard 
fineness and weight of coins. 

The nickel cent contains 88 parts of copper and 12 of nickel. 

The new bronze cent contains 95 parts of copper and 5 of tin and zinc. 

Pure gold, 23.22 grains = $1, or $20.67.183 = 1 ounce. 

Pure silver, 357.03 grains == $1, or $1.36.166 — 1 ounce. 

Silver coins of less value than one dollar are issued at the rate of 384 grains to 
the dollar. t 

Standard alloyed gold = $18.60.465, and silver = $1.22.5 per ounce. 


Gold coins. Grains. Silver coins. Grains. 

Double eagle,. . 516. One dollar, . . 412.5 

Eagle, . . . 258. Fifty cents,. . 192. 

Dollar, . . . 25.8 Twenty-five cents, 96. 

For silver and gold tables see pages 000. 


Copper coins. 

Cent (old), , 
Cent (new), 
Cent(bronze), 


Grains. 
. 168. 

72. 

. 48. 


WEIGHT AND FINENESS OF DIFFERENT COINS, AND 
THEIR VALUE IN AMERICAN MONEY. 


Country. 

Austria,. 

Baden,. . 
Belgium, 
Brazil, . . 
Canada, . 
China, . . 

Chili,. . 

Denmark, 
England, 
East Indies 
France, . 
Greece, . 
Hamburg, 

Holland, . 
Italy, 
Mexico, . 

Norway,. 
Peru, . . 

Portugal, 

Prussia, . 
Rome, 

Russia, 
Spain, . 

Sweden, . 
Turkey, . 


Piece and Divisions. 

f Crown, .... 

{ Florins. 

Ducat, .... 

25 Francs, .... 
2000 Reis, . . . 

20 Cents, 1851, . 

Tael, .... 
f 10 Pesos, 1855, . . . 

( 1 Peso, 1854-6, . 

2 Rix dollars, 

Pound sterling = 20 shillings 
Company’s Rupee,. 
Napoleon, 20 Francs,. 

20 Drachms, . 

Rix Dollar, 

f Ducat, .... 
(Florin, .... 

20 Lire, .... 
f Doubloon = 8 Escudos, 

(Peso = 8 Reals, . . 

2 Rigsdaler, . . , 

1 Sol = 100 Centavos, . 
f Corona (Crown), 1838, 

(1000 Reis, . . . 

Thaler, .... 
2.5 Scudi — 250 Bajochi, 
f Imperial = 5 Roubles, 

( Rouble silver = 100 Copecks 
(100 Reals, 

1 80 Reals = 4 Dollars, 

J Ducat, .... 

( Rix Dollar = 100 Ore, 
Piastres, 1845, . 


Weight. 

Grains. 

171.36 

190.56 

47.5 
121.92 
393.6 

96.0 

236.16 
384.48 
444.96 
123.21 
180. 

99.5 

88.8 

450. 

53.75 

50. 

99.36 

416.4 

415.68 

444.96 

385.82 

147.84 

45.6 
268.46 

67.2 

100.8 

320.16 
128.64 

103.2 
53. 

112.3 

110.88 


Fineness 
in 1000. 

900. 

900. 

987. 

899. 

918.5 
925. 

900, 

900.5 
877. 

916.5 
892. 
898. 

900. 
860. 
982. 
787. 
898. 

870.5 

901. 
877. 
900. 
912. 
912. 
900. 
900. 
916. 
875. 
896. 

869.5 
979. 
873. 
900. 


Fine 

metal. 

Grains. 

154.22 

171.5 

46.9 

109.6 

361.5 
88.8 

2*12.5 

346.2 

390.2 
112.9 

16.5 

87.4 

80.9 

397.5 
52.77 
39.32 
89.22 

362.5 

374.5 

390.2 
347.24 

134.8 

41.6 

243.6 

60.5 
92.3 

286.8 

115.2 
89.73 

51.9 
97.15 
99.79 


United 

States. 

$ Cts. 

6.64.19 

0.48.63 

2.00.70 

4.72.03 

1.02.53 

0.18.87 

1.43.00 

9.15.35 

0.98.17 

1.10.65 
4.86.34 

5.10.49 
3.85.00 

3.44.29 

1.17.66 
2.29.7 

1.69.30 
3.84.26 

15.61.05 

1.06.20 

1.10.65 
0.95.41 

5.80.66 
1.18.00 
0.72.89 
2.60.47 
3.97.64 
0.79.44 
4.96.39 
3.86.44 

2.23.50 
0.26.10 
4.36.93 















Foreign Money. 


THE CURRENCY OF DIFFERENT COUNTRIES COMPARED 
WITH ENGLISH AND AMERICAN MONEY. 


Engl’nd. 

France. 

Belgi’m. 

Sw’land. 

Prussia. 

Austria, 
(in notes.) 

Sweden. 

Ger¬ 

many. 

Russia, 
(in paper) 

Ham¬ 

burg. 

U. S. 

£ 

s. 

d. 

Frs. 

Cts. 

Th. 

Sgr. 

Pf. 

FI. 

Kr. 

Rix. Ore. 

FI. 

Kr. 

Rbl. 

Kop. 

Mrk 

Sell. 

$ Cts. 

0 

0 

1 

0 

10* 

0 

0 

10 

0 

5 

0.07 

0 

3 

0 

3 

0 

0 

0.02 

0 

0 

2 

0 

21 

0 

1 

8 

0 

10 

0.14 

0 

6 

0 

5 

0 

2 

0.04 

0 

0 

3 

0 

32 

0 

2 

6 

0 

16 

0.21 

0 

9 

0 

8 

0 

2* 

0.06 

0 

0 

4 

0 

42 

0 

3 

4 

0 

21* 

0.28 

0 

12 

0 

12 

0 

2* 

0.08 

0 

0 

5 

0 

53 

0 

4 

2 

0 

27 

0.36 

0 

15 

0 

16 

0 

4* 

0.10 

0 

0 

6 

0 

64 

0 

5 

1 

0 

31 

0.44 

0 

18 

0 

19 

0 

5* 

0.12 

0 

0 

7 

0 

74 

0 

5 

11 

0 

364 

0.51 

0 

21 

0 

22 

0 

6* 

0.14 

0 

0 

8 

0 

85 

0 

6 

10 

0 

42* 

0.59 

0 

24 

0 

26 

0 

7 

0.16 

0 

0 

9 

0 

96 

0 

7 

7 

0 

47* 

0.66 

0 

27 

0 

27 

0 

8 

0.18 

0 

0 

10 

1 

6 

0 

8 

6 

0 

53 

0.73 

0 

30 

0 

33 

0 

8# 

0.20 

0 

0 

11 

1 

16 

0 

9 

5 

0 

57* 

0.80 

0 

34 

0 

36 

0 

9$ 

0.22 

0 

1 

0 

1 

27 

0 

10 

3 

0 

62 

0.89 

0 

36 

0 

39 

0 

11 

024 

0 

o 

0 

2 

55 

0 

20 

6 

1 

25 

1.78 

1 

13 

0 

79 

1 

6 

0.48 

0 

3 

0 

3 

82 

1 

0 

9 

1 

87 

2.67 

1 

49 

1 

18 

2 

1 

0.72 

0 

4 

0 

5 

10 

1 

10 

11 

2 

50 

3.56 

2 

24 

1 

58 

2 

12 

0.96 

0 

5 

0 

6 

36 

1 

21 

3 

3 

12 

4.45 

2 

59 

1 

97 

3 

7 

1.21 

0 

6 

0 

7 

64 

2 

1 

6 

3 

74 

5.34 

3 

38 

2 

37 

4 

2 

1.45 

0 

7 

0 

8 

92 

2 

11 

9 

4 

36 

6.23 

4 

12 

2 

77 

4 

12 

1.69 

0 

8 

6 

10 

20 

2 

22 

0 

4 

95 

7.12 

4 

47 

3 

18 

5 

7 

1.93 

0 

9 

0 

11 

46 

3 

2 

0 

5 

58 

8.09 

5 

22 

3 

58 

6 

2* 

2.18 

0 10 

0 

12 

72 

3 

12 

4 

6 

25 

8.90 

5 

58 

3 

94 

6 

13* 

2.42 

0 11 

0 

13 

99 

3 

22 

6 

6 

87 

9.79 

6 

34 

4 

38 

7 

8* 

2.66 

0 12 

0 

15 

27 

4 

2 

9 

7 

49 

10.68 

7 

11 

4 

75 

8 

3* 

2.90 

0 

13 

0 

16 

55 

4 

13 

0 

8 

12 

11.57 

7 

46 

5 

15 

8 

14* 

3.14 

0 

14 

0 

17 

84 

4 

23 

3 

8 

75 

12.66 

8 

24 

5 

55 

9 

9* 

3.39 

0 

15 

0 

19 

8 

5 

3 

5 

9 

37 

13.45 

8 

57 

5 

96 

10 

4* 

3.63 

0 

16 

0 

20 

40 

5 

13 

8 

10 

0 

14.24 

9 

33 

6 

35 

10 

15* 

3.87 

0 17 

0 

21 

66 

5 

23 

11 

10 

65 

15.13 

10 

9 

6 

74 

11 

10* 

3.12 

0 18 

0 

22 

92 

6 

4 

2 

11 

28 

16.02 

10 

46 

7 

14 

12 

5* 

4.36 

0 

19 

0 

24 

18 

6 

14 

4 

11 

88 

17.01 

11 

21 

7 

44 

13 

0* 

4.60 

1 

0 

0 

25 

45 

6 

24 

6 

12 

50 

17.80 

11 

57 

7 

88 

13 

9 

4.84 

o 

0 

0 

50 

90 

13 

19 

0 

25 

0 

35.60 

23 

54 

15 

77 

27 

2 

9.68 

3 

0 

0 

76 

35 

20 

13 

6 

37 

50 

53.40 

35 

51 

23 

t55 

40 

11 

14.52 

4 

0 

0 

101 

80 

27 

8 

0 

50 

0 

71.20 

47 

48 

31 

54 

54 

4 

17.36 

5 

0 

0 

127 

25 

34 

3 

0 

62 

50 

89.00 

59 

46 

39 

42 

67 

11 

24.20 

6 

0 

0 

152 

70 

40 

27 

6 

75 

0 

106.80 

71 

42 

47 

31 

81 

4 

29.04 

7 

0 

0 

178 

15 

47 

22 

6 

87 

50 

124.60 

83 

39 

55 

20 

94 

13 

33.88 

8 

0 

0 

202 

60 

54 

16 

6 

100 

0 

142.40 

95 

36 

63 

9 

108 

6 

38.72 

9 

0 

0 

229 

5 

61 

11 

6 

112 

50 

160.20 

107 

34 

70 

96 

121 

15 

43.56 

10 

0 

0 

254 

50 

68 

6 

0 

125 

0 

178.00 

119 

30 

78 

84 

135 

8 

48.40 


The mark of Finland is equal to the French franc. 


Carat. 

DIAMOND. 

Grain. 

Parts. 

Grains (Troy). 

1 . 

4. 

64 

3.2 

0.25 

1 . 

16 

0.8 

0.015625 

0.0625 

1 

0 05 

0.3125 

12.5 

20 

1 . 















































































35 


Rule Measure. 


Conversion of Indies ancl Eiglitlis into Decimals of a Foot. 


Fractions of an Inch. 


Inches. 

0 

1 

8 

l 

¥ 

3 

8 

1 

2 

5 

8 

t 

7 

8 

0 

.0000 

.01041 

.02083 

.03125 

.04166 

.05208 

.0625 

.07291 

1 

.08333 

.09375 

.10416 

.11458 

.125 

.13541 

.14588 

.15639 

2 

.16666 

.17707 

.1875 

.19792 

.20832 

.21873 

.22914 

.23965 

3 

.25 

.26041 

.270 

.28125 

.29166 

.30208 

.3125 

.32291 

4 

.33333 

.34375 

.35416 

.364 

.375 

.38541 

.39588 

.40639 

5 

.41666 

.42707 

.437 

.44792 

.45832 

.46873 

.47914 

.48965 

6 

.5 

.51041 

.520 

.53125 

.54166 

.55208 

.5625 

.57291 

7 

.58333 

.59375 

.60416 

.614 

.625 

.63541 

.64588 

.65639 

8 

.66666 

.67707 

.685 

.69792 

.70832 

.71773 

.72914 

.73965 

9 

.75 

.76041 

.770 

.78125 

.79169 

.80208 

.8425 

.82291 

10 

.83333 

.84375 

.85416 

.864 

.875 

.88541 

.89588 

.90639 

11 

.91666 

.92707 

.937 

.94792 

.95832 

.96873 

.97914 

.98965 

12 

1 foot. 

foot. 

foot. 

foot. 

foot. 

foot. 

foot. 

foot. 


T v in. = 0.005208 ft.; in. = 0.00265 ft.; fa in. = 0.001375 ft. 


Angle Measurement toy tlie Opening of a Two-foot Rule. 


Opening 

Rule, 

0 

1 

¥ 

1 

1 

Fi 

(.ACTIONS 

3 

8 

3F AN lN< 
1 
¥ 

:h. 

5 

8 

3 

T 

¥ 

Inch’s. 

o 

r 

O ' 

o 

/ 

o r 

o t 

o t 

o / 

o / 

0 

00 

00 

0 36 

1 

12 

1 47 

2 23 

2 59 

3 35 

4 11 

1 

4 

46 

5 22 

5 

59 

6 34 

7 10 

7 46 

8 22 

8 58 

2 

9 

34 

10 10 

10 

46 

11 22 

11 58 

12 34 

13 10 

13 46 

3 

14 

22 

14 58 

15 

34 

16 10 

16 46 

17 22 

17 59 

18 35 

4 

19 

12 

19 48 

20 

24 

21 0 

21 37 

22 13 

22 50 

23 27 

5 

24 

3 

24 39 

25 

16 

25 53 

26 30 

27 7 

27 44 

28 21 

6 

28 

58 

29 35 

30 

12 

30 49 

31 26 

32 3 

32 40 

33 17 

7 

33 

54 

34 33 

35 

8 

35 46 

36 25 

37 3 

37 40 

38 18 

8 

38 

56 

39 34 

40 

12 

40 50 

41 29 

42 7 

42 46 

43 24 

9 

44 

4 

44 42 

45 

21 

45 59 

46 38 

47 17 

47 56 

48 35 

10 

49 

15 

49 54 

50 

34 

51 13 

51 53 

52 33 

53 13 

53 53 

11 

54 

34 

55 14 

55 

55 

56 35 

57 16 

57 57 

58 38 

59 19 

12 

60 

0 

60 41 

61 

23 

62 5 

62 47 

63 28 

64 10 

64 52 

13 

65 

35 

66 18 

67 

1 

67 44 

68 28 

69 12 

69 55 

70 38 

14 

71 

20 

72 6 

72 

51 

73 35 

74 21 

75 6 

75 51 

76 36 

15 

77 

20 

78 8 

78 

54 

79 40 

80.27 

81 14 

82 1 

82 49 

16 

83 

38 

84 26 

85 

14 

86 3 

86 52 

87 41 

88 31 

89 21 

17 

90 

12 

91 3 

91 

55 

92 41 

93 39 

94 31 

95 34 

96 17 

18 

97 

11 

98 5 

99 

0 

99 55 

100 51 

101 47 

102 44 

103 42 

19 

104 

40 

105 39 

105 

39 

107 40 

108 41 

109 43 

110 46 

111 49 

20 

112 

53 

113 58 

115 

4 

116 11 

117 20 

118 30 

149 41 

120 53 


Conversion of Vulgar Fractions into Decimals. 


Fract’ns. 

Decimals. 

Fract ns. 

Decimals. 

Fract’ns. 

Decimals. 

Fract’ns. 

Decimals. 

1:2 

.5 

1:16 

.0625 

1:32 

.03125 

1:64 

.015625 

1:3 

.33333 

3:16 

.1875 

3:32 

.09375 

3:64 

.046875 

2:3 

.66666 

5:16 

.3125 

5:32 

.15625 

5:64 

.078125 

1:4 

.25 

7:16 

.4375 

7:32 

.21875 

7:64 

.109375 

3:4 

.75 

9:16 

.5625 

9:32 

.28125 

9:64 

.140625 

1:5 

.2 

11:16 

.6875 

11:32 

.34375 

11:64 

.171875 

3:5 

.6 

13:16 

.8125 

13:32 

.40625 

15:64 

.234375 

1:6 

.16666 

15:16 

.9375 

15:32 

.46875 

19:64 

.296875 

5:6 

.83333 

1:24 

.04166 

17:32 

.53125 

23:64 

.359375 

1:8 

.125 

5:24 

.20833 

19:32 

.59375 

27:64 

.421875 

3:8 

.375 

7:24 

.29166 

21:32 

.65625 

31:64 

.484375 

5:8 

.625 

11:24 

.45833 

23:32 

.71S75 

35:64 

.546875 

7:8 

.875 

13:24 

.54166 

25:32 

.78125 

39: 64 

.609375 

5:12 

.41666 

17:24 

.70833 

27:32 

.84375 

43:64 

.671875 

7:12 

.58333 

19:24 

.79166 

39:32 

.90625 

57:64 

.891625 

11:12 

.925 

23:24 

.95833 

31:32 

.96S75 

61:64 

.953125 







































































r 


Metrical System. 


37 


To Determine an Angle l»y the Aid of a Two-foot Rule. 

b = opening of the rule in inches; 
v — angle formed by the rule; 

Sin. = — ; and b — 24 sin. 4u. 

24 

Example 1. How much (b — ?) must a two-foot rule he opened to form an angle 
of 48° 40'? 

b = 24 + sin. 24° 20' = 24 X 0.412 = 9.888 inches. 

Example 2. A two-foot rule is opened to b = 8 inches. Required the angle 
formed by the rule. 

Sin. = — = 0.3333 = sin. 19° 30', and v = 39°. 

24 

THE FRENCH METRICAL SYSTEM. 

The French units of weight, measure and coin are arranged into a perfect deci¬ 
mal system, except those of time and the circle. The division and multiplication 
of the units are expressed by Latin and Greek names, as follow: 


Latin , Division. 

Milli = 1000th of the unit. 

Centi = 100th of the unit. 

Deci = 10th of the unit. 

Metre, Litre, Stere, Are, Franc, Gramme. 


Greek , Multiplication. 
Deca = 10 times the unit. 
Hecato = 100 times the unit. 
Kilio = 1000 times the unit. 
Myrio = 10000 times the unit. 


French Measure of Length. 


Millimetre 
Centimetre 
Decimetre 
Metre (unit) 

Sea mile or j _ 
knot j 

1 Kilometre = 0.541343 sea miles. 


= 0.03937079 inches. 
= 0.3937079 inches. 
— 3.937079 inches. 
= 39.37079 inches. 


1.8472 kilometre. 


1 Metre (unit) 
1 Decametre 
1 Hectometre 
1 Kilometre 

1 Statute mile 
1 Kilometre 


= 3.280899 feet. 

= 32.80899 feet. 

= 328.0899 feet 
= 3280.899 ft. •- * 0.62138 
mile. 

= 1.609315 kilometres. 

= 49.7106 chains. 


French Measure of Surface. 


1 Sq. metre = 10.7643 square feet. 
1 Are = 100 square metres. 

1 Decare = 10 ares. 

1 Hectare = 100 ares. 


1 Are = 1076.43 square feet. 

1 Decare = 107.643 square feet.* 
1 Hectare == 2.47114 Eng. acres. 

1 Sq. mile = 258.989 hectares. 


French Measure of Volume. 


1 Stere (cubic 1 = 1Q decasteres . 


metre) 

1 Stere 
1 Litre 
1 Decistere 


1 Ton 


J 


= 1000 litres. 

= 1 cubic decimetre. 
== 3.53166 cubic feet. 


1 Stere = 35.3166 Eng. cubic feet. 

1 Litre = 61.0271 Eng. cubic inches 

1 Gallon = 3.7852 litres. 

1 Decistere = 2.84 bushels. 


dis- 


1 cubic metre 
tilled water. 

1000 kilogrammes. 
1000 grammes. 

100 grammes. 

10 grammes. 

1 cubic centimetre 
distilled water. 

1 French ton = 0.984274 Eng. tons. 


1 Ton 

1 Kilogramme 
1 Hectogramme 
1 Decagramme 
1 Gramme 


French Measure of Weight 

Gramme = 
Decigramme = 
Centigramme = 
Kilogramme = 


10 decigrammes. 

10 centigrammes. 

10 milligrammes. 
2.2047 pounds avoir¬ 
dupois. 

1 Eng. pound = 0.45358 kilogrammes 


Gramme 
English ton 


15.43315 grains troy. 
1.01598 French tons 


French Coin. 

1 Franc 100 centimes — 19.06 cents of an American dollar. 















48 


Feet and Metres. 


Conversion of English. Inches into Centimetres. 

Inch's 

0 

I 

2 

3 

4 r 

5 

6 

7 

8 

9 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Ct.mt. 

0.000 

25.40 

50.80 

76.20 

101.60 

127.00 

152.40 

177.80 

203.20 

228.60 

254.00 

Ct.mt. 

2.540 

27.94 

53.34 

78.74 

104.14 

129.54 
154.94 
180.34 
205.74 

231.14 

256.54 

Ct.mt. 

5.080 

30.48 

55.88 

81.28 

106.68 

132.08 

157.48 

182.88 

208.28 

233.68 

259.08 

Ct.mt. 

7.620 

33.02 

58.42 

83.82 

109.22 

134.62 
160.02 
185.42 
210.82 

236.22 

261.62 

Ct.mt. 

10.16 

35.56 

60.96 

86.36 

111.76 

137.16 
162.56 
187.96 
213.36 

238.76 

264.16 

Ct.mt. 

12.70 

38.10 

63.50 

88.90 

114.30 

139.70 
165.10 
190.50 
215.90 

241.30 

266.70 

Ct.mt. 

15.24 

40.64 

66.04 

91.44 

116.84 

142.24 
167.64 
193.04 
218.44 

243.84 

269.24 

Ct.mt. 

17.78 

43.18 

68.58 

93.98 

119.38 

144.78 
170.18 
195.58 
220.98 

246.38 

271.78 

Ct.mt. 

20.32 

45.72 

71.12 

96.52 

121.92 

147.32 
172.72 
198.12 
223.52 

248.92 

274.32 

Ct.mt . 
22.86 
48.26 
73.66 
99.06 

124.46 
149.86 
175.26 
200.96 
226.06 

251.46 
276.85 

Conversion of Centimetres into English Inches. 

Ct. mt. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 I 

Inches. 

0.000 

3.937 

7.874 

11.811 

15.748 

19.685 

23.622 

27.560 

31.497 

35.434 

39.370 

Inches. 
0.394 
4.331 
8.268 
12.205 
16.142 
20 079 
24.016 
27.953 
31.890 
35.827 
39.764 

Inches. 

0.787 

4.742 

8.662 

12.599 

16.536 

20.473 

24.410 

28.347 

32.284 

36.221 

40.158 

Inches.- 

1.181 

5.118 

9.055 

12.992 

16.929 

20.867 

24.804 

28.741 

32.678 

36.615 

40.552 

Inches. 

1.575 

5.512 

9.449 

13.386 

17.323 

21.260 

25.197 

29.134 

33.071 

37.009 

40.945 

Inches. 

1.969 

5.906 

9.843 

13.780 

17.717 

21.654 

25.591 

29.528 

33.465 

37.402 

41.339 

Inches. 

2.362 

6.299 

10.236 

14.173 

18.111 

22.048 

25.985 

29.922 

33.859 

37.796 

41.733 

Inches. 

2.756 

6.693 

10.630 

14.567 

18.504 

22.441 

20.378 

30.316 

34.253 

38.190 

42.126 

Inches. 

3.150 

7.087 

11.024 

14.961 

18.898 

22.835 

26.772 

30.709 

34.646 

38.583 

42.520 

Inches. 

3.543 

7.480 

11.418 

15.355 

19.292 

23.229 

27.166 

31.103 

35.040. 

38.977' 

42.914 

Conversion of English Feet into Metres. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Met. 

0.000 
3.0479 
*6.0359 
9.1438 
12.192 
15.239 
18 287 
21.335 
24.383 
27.431 
30.479 

Met. 

0.304S 

3.3527 

6.4006 

9.4486 

12.496 

15.544 

18.592 

21.640 

24.688 

27.736 

30.784 

Met. 

0.6096 

3.6575 

6.7055 

9.7534 

12.801 

15.849 

18.897 

21.945 

24.993 

28.041 

31.0S9 

Met. 

0.9144 

3.9623 

7.0102 

10.058 

13.106 

16.154 

19.202 

22.250 

25.298 

28.346 

31.394 

Met. 

1.2192 

4.2671 

7.3150 

10.363 

13.411 

16.459 

19.507 

22.555 

25.602 

28.651 

31.698 

Met. 

1.5239 

4.5719 

7.6198 

10.668 

13.716 

10.763 

19.811 

22.859 

25.907 

28.955 

32.003 

Met. 

1.8287 

4.8767 

7.9246 

10.972 

14.020 

17.068 

20.116 

23.164 

26.212 

29.260 

32.308 

Met. 

2.1335 

5.1815 

8.2294 

11.277 

14.325 

17.373 

20.421 

23.469 

26.517 

29.565 

32.613 

Met. 

2.4383 

5.4863 

8.5342 

11.582 

14.630 

17.678 

20.726 

23.774 

26.822 

29.870 

32.918 

Met. 

2.7431 

5.7911 

8.8390 

11.887 

14.935 

17.983 

21.031 

24.079 

27.126 

30.174 

33.222 

Conversion of Metres into English Feet. 


Metres. 

0 

1 

2 

3 

4: 

5 

6 

7 

8 

9 


Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5618 

9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.371 

42.651 

45.932 

49.213 

52.494 

55.775 

59.056 

62.337 

20 

65.618 

68.899 

72.179 

75 461 

78.741 

82.022 

85.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99 

108.27 

111.55 

114.83 

118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80 

141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

160.76 

50 

164.04 

107.33 

170.61 

173.S9 

177.17 

180.45 

183.73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42 

206.70 

209.98 

213 26 

216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22 

239.51 

242.79 

246.07 

249.35 

252.63 

255.91 

259.19 

80 

262.47 

265.75 

269.03 

272.31 

275.60 

278.88 

282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

391.84 

305.12 

308.40 

311.69 

314.97 

318.25 

321.53 

324.81 

100 

328.09 

331.37 

334.65 

337.93 

341.21 

344.49 

347.7S 

351.06 

354.34 

357.62 





































































































Miles and Kilometres 


39 


Conversion of English Statute-miles into Kilometres. 

Miles. 

0 

1 

2 

3 

4r 

5 

6 

7 

8 

9 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Kilom. 

0.0000 

16.093 

32.186 

48.279 

64.372 

80.465 

96.558 

112.65 

128.74 

144.85 

160.93 

Kilom. 

1.6093 

17.702 

33.795 

49.888 

65.981 

82.074 

98.167 

114.26 

130.35 

146.44 

162.53 

Kilom. 

3.2186 

19.312 

35.405 

51.498 

67.591 

83.684 

99.777 

115.87 

131.96 

148.05 

164.14 

Kilom. 
4.8279 
20.921 
37.014 
53.107 
69.200 
85.293 
101.39 
117.48 
133.57 
149.66 
165.75 

Kilom. 

6.4372 

22.530 

38.623 

54.716 

70.809 

86.902 

102.99 

119.08 

135.17 

151.26 

167.35 

Kilom. 

8.0465 

24.139 

40.232 

56.325 

72.418 

88.511 

104.60 

120.69 

136.78 

152.87 

168.96 

Kilom. 

9.6558 

25.749 

41.842 

57.935 

74.028 

90.121 

106.21 

122.30 

138.39 

154.48 

170.57 

Kilom. 

11.2652 

27.358 

43.451 

59.544 

75.637 

91.730 

107.82 

123.91 

140.00 

156.09 

172.18 

Kilom. 

12.8745 

28.967 

45.060 

61.153 

77.246 

93.339 

109.43 

125.52 

141.61 

157.70 

173.79 

Kilom. 

14.4848 

30.577 

46.670 

62.763 

78.856 

94.949 

111.04 

127.13 

143.22 

159.31 

175.40 

Conversion of Kilometres into Englisli Statnte-miles. 

Kilom. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Miles. 

0.0000 

6.2138 

12.427 

18.641 

24.855 

31.069 

37.282 

43.497 

49.711 

55.924 

62.138 

Miles. 

0.6214 

6.8352 

13.049 

19.263 

25.477 

31.690 

37.904 

44.118 

50.332 

56.545 

62.759 

Miles. 

1.2427 

7.4565 

13.670 

19.884 

26.098 

32.311 

38.525 

44.739 

50.95S 

57.166 

63.380 

Miles. 

1.8641 
8.0780 
14.292 
20.506 
26.720 
32.933 
39.147 
45.361 
51.575 
57.788 
1 64.002 

Miles. 

2.4855 

8.6994 

14.913 

21.127 

27.341 

33.554 

39.768 

45.982 

52.196 

58.409 

64.623 

Miles. 

3.1069 

9.3208 

15.534 

21.748 

27.962 

34.175 

40.389 

46.603 

52.817 

59.030 

65.244 

Miles. 

3.7282 

9.9421 

16.156 

22.370 

28.584 

34.797 

41.011 

47.225 

53.439 

59.652 

65.866 

Miles. 

4.3497 

10.562 

16.776 

22.990 

29.204 

35.417 

41.631 

47.845 

54.059 

60.272 

6C.486 

Miles. 

4.9711 

11.185 

17.399 

23.613 

29.827 

36.040 

42.254 

48.468 

54.682 

60.895 

67.109 

Miles. 

5.5924 

11.805 

18.019 

24.233 

30.447 

36.660 

42.874 

49.088 

55.302 

61.515 

67.729 

Conversion of Sea-miles, Knots or Minutes into Kilometres. 


Knots. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom, 

Kilom. 

0 

0.0000 

1.8472 

3.6944 

5.5416 

7.3888 

9.2361 

11.083 

12.930 

14.777 

16.625 

10 

18.472 

20.319 

22.166 

24.013 

25.861 

27.708 

29.555 

31.402 

33.249 

35.097 

20 

36.944 

38.791 

40.638 

42.485 

44.333 

46.180 

48.027 

49.874 

51.721 

53.569 

30 

55.416 

57.263 

59.110 

60.957 

62.805 

64.652 

66.499 

68.346 

70.193 

72.041 

40 

73.888 

75.735 

77.582 

79.429 

81.277 

83124 

84.971 

86.818 

88.665 

90.513 

50 

92.361 

94 207 

96.054 

97.901 

99.749 

101.59 

103.44 

105.29 

107.14 

108.98 

60 

110.83 

112.68 

114.53 

116.37 

118.22 

120.06 

121.91 

123.76 

125.61 

127.45 

70 

129.30 

131.15 

133.00 

134.84 

136.70 

138.54 

140.39 

142.24 

144.09 

145.94 

80 

147.77 

149.62 

151.47 

153.31 

155.18 

157.02 

158.87 

160.72 

162.57 

164.43 

90 

166.25 

168.09 

169.94 

171.78 

173.65 

175.49 

177.34 

179.19 

181.04 

182.90 

100 

184.72 

186.56 

188.41 

190.25 

192.12 

193.96 

195.81 

198.66 

199.51 

201.37 

Conversion of Kilometres into Sea-miles, Knots or Minutes. 

Kilom. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 


Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

0 

0.0000 

0.5413 

1.0827 

1.6240 

2.1653 

2.7066 

3.2480 

3.7894 

4.3307 

4.8721 

10 

5.4134 

5.9547 

6.4961 

7.0374 

7.5787 

8.1200 

8.6614 

9.2028 

9.7441 

10.285 

20 

10.827 

11.368 

11.909 

12.451 

12.992 

13.533 

14.075 

14.616 

15.157 

15.702 

30 

16.24 

16.781 

17.322 

17.864 

18.406 

18.946 

19.488 

20.029 

20.570 

21.115 

40 

21.653 

22.194 

22.735 

23.277 

23.819 

24.359 

24.901 

25.442 

25.983 

26.528 

50 

27.066 

27.607 

28.148 

28.090 

29.232 

29.772 

30.314 

30.855 

31.396 

31.941 

60 

32.480 

33.020 

33.561 

34.103 

34.645 

. 35.185 

35.727 

36.268 

36.809 

37.364 

70 

37.894 

38.433 

38.974 

39.516 

40.058 

40.598 

41.140 

41.681 

42.222 

42.777 

80 

43.307 

43.846 

44.387 

44.929 

45.471 

46.011 

46.553 

47.094 

47.635 

48.190 

90 

48.721 

49.259 

49.800 

50.342 

50.884 

51.424 

51.966 

52.507 

53.048 

54.603 

100 

54.134 

54.672 

55.213 

55.755 

56.297 

56.837 

57.379 

57.920 

58.461 

60.016 





































































































40 


Foot-measures and Pounds, 


Comparison between Foot-measnres of Different Nations. 

Linear Feet. 

English. 

Metre. 

Prussia. 

Saxony. 

Baden. 

Austria. 

Hanover 

Sweden. 

1 

0.3048 

0.9711 

1.0763 

1.0160 

0.9642 

1.0435 

1.0265 

3.2809 

1 

3.1862 

3.5312 

3.3333 

3.1634 

3.4235 

3.3678 

1.0297 

0.3138 

1 

1.1083 

1.0462 

0.9929 

1.0745 

1.0572 

0.9291 

0.2832 

0.9023 

1 

0.9440 

0.8959 

0.9695 

0.9538 

0.9843 

0.3000 

0.9559 

1.0594 

1 

0.9490 

1.0271 

1.0164 

1.0371 

0.3161 

1.0072 

1.1163 

1.0537 

1 

1.0822 

1.0963 

0.9583 

0.2921 

0.9307 

1.0314 

0.9736 

0.9240 

1 

0.9838 

0.9741 

0.2969 

0.9459 

1.0484 

0.9838 

0.9122 

1.0165 

1 

Square Feet. 

1 

0.0929 

0.9431 

1.1584 

1.0322 

0.9297 

1.0888 

1.0537 

10.764 

1 

10.152 

12.469 

11.111 

10.007 

11.721 

11.342 

1.0603 

0.0985 

1 

1.2283 

1.0945 

0.9858 

1.1545 

1.1130 

0.8603 

0.0802 

0.8141 

1 

0.8911 

0.8026 

0.9400 

0.9097 

0.9688 

0.0900 

0.9137 

1.1222 

1 

0.9007 

1.0549 

1.0330 

1.0756 

0.0999 

1.0144 

1.2460 

1.1103 

1 

1.1712 

1.2019 

0.9184 

0.0853 

0.8661 

1.0639 

0.9480 

0.8538 

1 

0.9679 

0.9489 

0.0881 

0.8947 

1.0941 

0.9679 

0.8321 

1.0331 

1 

Cubic Feet. 

1 

0.0283 

0.9159 

1.2468 

1.0487 

0.8964 

1.1362 

1.1018 

35.316 

1 

32.346 

44.032 

37.037 

31.658 

40.126 

38.198 

1.0918 

0.0309 

l 

1.3613 

1.1450 

0.9787 

1.2405 

1.1816 

0.8021 

0.0227 

0.7346 

1 

0.8411 

0.7190 

0.9113 

0.8677 

0.9535 

0.0270 

0.8733 

1.1889 

1 

0.8548 

1.0834 

1.0501 

1.0756 

0.0999 

1.0144 

1.2460 

1.1103 

1 

1.1712 

1.3176 

0.8801 

0.0249 

0.8061 

1.0973 

0.9230 

0.7890 

1 

0.9522 

0.9243 

0.0262 

0.8483 

1.1444 

0.9522 

0.7590 

1.0501 

1 

Conversion of Pounds of Different Nations. 

Eng. av. 

Kilogram. 

Prussia. 

Austria. 

Spain. 

Hanover 

Russia. 

Sweden. 

1 

0.4536 

0.9072 

0.8110 

0.9839 

0.9320 

1.1076 

1.0664 

2.2046 

1 

2.0000 

1.7857 

2.1692 

1.9842 

2.4419 

2.3511 

1.1023 

0.5000 

1 

0.8929 

1.0857 

1.0271 

1.2209 

1.1755 

1.2346 

0.5600 

1.1200 

1 

1.2132 

1.1490 

1.3675 

1.3166 

1.0164 

0.4610 

0.9211 

0.8243 

1 

0.9470 

1.1257 

1.0839 

1.0730 

0.4696 

0.9752 

0.8596 

1.0557 

1 

1.1884 

1.1442 

0.9028 

0.4095 

0.8190 

0.7313 

0.8883 

0.8414 

1 

0.9628 

0.9377 

0.4253 

0.8508 

0.7595 

0.9226 

0.8738 

1.0386 

1 


Ancient Measures of Length. 


Scripture. 

Feet. 

Inches. 

Hebrew. 

Feet. 

Inches 

Digit, .... 

• • • 

0.912 

Cubit, .... 

1 

9.868 

Palm = 4 Digits, . 

• • • 

3.648 

Sabbath day’s journey, 

3648 

• • • 

Span = 3 Palms, . 

• • • 

10.94 

Mile =4000 Cubits, 

7296 


Cubit = 2 Spans, 

1 

9.888 

Day’s journey = 33.164 mi. 

• • • 


Fathom = 3.46 Cubits,. 

7 

3.552 

Sacred Cubit, . . 

2 

0.24 

Egyptian. Finger, 

• • • 

.7374 

Roman. 



Nahud Cubit, . 

1 

5.71 

Digit, .... 

• • • 

.7257 

Royal Cubit, . . 

1 

8.66 

Uncia (Inch), 

• • • 

.967 




Pes (foot) = 12 Uncias, . 

• • • 

11.60 

Digit, .... 
Pous = 16 Digits, 

Cubit, .... 


0.754 

.0875 

1.598 

Cubit = 24 Digits, 

1 

5.406 

1 

1 

Passus = 3.33 Cubits, 
Millarium (mile), . 

4 

4842 

10.02 

Stadium, . 

604 

4.5 

Arabian. Foot, 

1 

1.14 

Mile = 8 Stadiums, 

4835 


Babylonian, Foot, 

1 

1.68 
































































Foreign Weights and Measures. 41 


Foreign Measures of Length Compared with American. 

Places. 

Measures. 

Inches . 

Places. 

Measures. 


Inches. 

Amsterdam, 

Foot. . 

11-14 

Malta, . . 

Foot, 


11-17 

Antwerp, . 

66 

• • • 

11-24 

Moscow, . 

(( 


13-17 

Bavaria, . 

66 

• • • 

11-42 

Naples, . . 

Pal mo, 


10-38 

Berlin, . . 

66 

• • • 

12-19 

Prussia, . 

Foot, . 


12-36 

Bremen, . 

66 

• • • 

11-38 

Persia, . . 

Arish, 


38-27 

Brussels, 

66 

• • • 

11-45 

Rhineland, 

Foot, . 


12-35 

China, . . 

“ mathematic, 

13-12 

Riga, . . 

66 


1079 

66 

“ builder’s. 

12-71 

Rome, 

66 

• • 


11-60 

• 

“ tradesman’s, 

13-32 

Russia, . . 

66 

• • 


13-75 

66 

• • 

“ surveyor’s . 

12-58 

Sardinia, 

Pal mo, 


9-78 

Copenhagen, 

66 

• • • 

12-35 

Sicily, . . 

66 


9-53 

Dresden, 

66 

11-14 

Spain, 

Foot, . 


11-03 

England, . 

66 

• • r • 

12-00 

66 

• a 

Toesas, 


66-72 

Florence, . 

Braccio. 

21-69 

66 

Palmo, 


8-64 

France, . 

Pied de Roi, . 

12-79 

Strasburg, 

Foot, 


11-39 

<6 

• • 

Metre, . 

39.381 

Sweden, . 

66 


11-69 

Geneva, . 

Foot, 

19-20 

Turin, 

66 


12-72 

Genoa, . . 

Pal mo, 

9-72 

Venice,. . 

66 


13-40 

Hamburg, 

Foot, 

11-29 

Vienna, . 

66 


12-45 

Hanover, . 

<( 

• s • 

11-45 

Zurich, . . 

66 


11-81 

Leipsic, . 

66 

11-11 

Utrecht, . 

66 

• 9 


10-74 

Lisbon, . . 

66 

• • • 

12-96 

Warsaw, . 

66 


1403 

66 

Pal mo, 

8-64 





Foreign Road Measures Compared with American 

• 

Places. 

Measures. 

Yards. 

Places. 

Measures. 


Yards. 

Arabia, . . 

Mile, . 

2148 

Hungary, . 

Mile, . 

# 

9113 

Bohemia, . 

66 

10137 

Ireland, . 

66 

• • • 


3038 

China, . . 

Li, . . . 

629 

Netherlands, 

66 


1093 

Denmark, 

Mile, 

8244 

Persia, . . 

Parasang, 


6086 

England, . 

“ statute, 

1760 

Poland, . 

Mile, long, . 

# 

8101 

66 

“ geographical. 

2025 

Portugal, . 

League, . 


6760 

Flanders, . 

66 

6869 

Prussia, . 

Mile, . 

. 

8468 

France, . 

League, marine, 

6075 

Rome, . . 

66 


2025 

66 

“ common,. 

4861 

Russia, . 

Verst, . 

# 

1167 

66 

“ post, 

4264 

Scotland, . 

Mile, 


1984 

Germany, . 

Mile, long, . 

10126 

Spain,. . 

League, common, 

7416 

Hamburg, 

66 

• • 

8244 

Sweden, 

Mile, . 

. 

11700 

Hanover, . 

66 

• • • 

11559 

Switzerland, 

66 


9153 

Holland,. 

66 

6395 

Turkey, . . 

Berri, . 

• 

1826 

Foreign Measures of Surface Compared with American. 

Places. 

Measures. 

Sq. Yds. 

Places. 

Measures. 


Sq. Yds. 

Amsterdam, 

Morgen, 

9722 

Portugal, . 

Geira, 

. 

6970 

Berlin, . . 

“ great, 

6786 

Prussia, . 

Morgen, 


3053 

It 

“ small, 

3054 

Rome, . . 

Pezza, 


3158 . 

Canary Isles, 

Fanegada, 

2422 

Russia, 

Dessetina, 


13066-6 

England, . 

Acre, . 

4840 

Scotland, . 

Acre,. 

. 

6150 

Geneva, . 

Arpent, 

6179 

Spain, . . 

Fanegada, 


5500 

Hamburg, . 

Morgen, . 

11545 

Sweden, 

Tuuneland, 

• 

5900 

Hanover, . 

66 

3100 

Switzerland, 

Faux, . 


7855 

Ireland,. . 

Acre, 

7840 

Vienna, . . 

Joch, 


6889 

Naples, 

Moggia, . 

3998 

Zurich, . 

Common acre, 

1 

3875-0 




































♦2 Foreign Weights and Measures. 


Foreign. Liquid Measures Compared with American. 

Places. 

Measures. 

Cub. In. 

Places. 

Measures. 

Cub. In. 

Amsterdam, . . 

Anker,. . . 

2331 

Naples,. . 

Wine Barille, 

2544 

U 

Stoop, . . 

146 

Li 

Oil Stajo, . 

1133 

Antwerp, . . . 

44 

194 

Oporto,. . 

Almude, . . 

1555 

Bordeaux, . . 

Barrique, . 

14033 

Rome, 

Wine Barille, 

2560 

Bremen, . . . 

Stubgens,. . 

194-5 

(( 

Oil 

2240 

Canaries, . . 

Arrobas, 

949 

« 

Boccali, . . 

80 

Constantinople, 

Almud, . . 

319 

Russia, . . 

Weddras, . 

752 

Copenhagen. 

Anker, . . 

2355 

U 

Kunkas. . . 

94 

Florence, . . . 

Oil Barille, . 

1946 

Scotland, . 

Pint, . . . 

103-5 

a 

Wine “ . 

2427 

Sicily, . 

Oil Caffiri, 

662 

France, . . 

Litre, . . . 

61-07 

Spain, . . 

Azumbras, . 

22-5 

Geneva, . . . 

Setier, . . 

2760 

a 

Quartillos, '. 

30-5 

Genoa, .... 

Wine Barille, 

4530 

Sweden, 

Eimer, . . 

4794 


Piute, . . . 

90-5 

U 

Kanna, . . 

159-57 

Hamburg, . . 

Stnbgen, . 

221 

Trieste, . , . 

Orne,. . . 

4007 

Hanover, . . 

44 

231 

Tripoli, . 

Mattari, . . 

1376 

Hungary,. . . 

Eimer, . . 

4474 

Tunis, . . 

Oil “ . . 

1157 

Leghorn, . . 

Oil Barille, . 

1942 

Venice, . 

Secchio, . . 

628 

Lisbon, . . . 

Almude, 

1040 

Vienna, 

Eimer, . . 

3452 

Malta, . . . 

Caffiri, . . 

1270 

LL 

Maas, . . . 

86-33 

Foreign Dry Measures Compared with American. 

Places. 

Measures. 

Cub. In. 

Places. 

Measures. 

Cub. In. 

Alexandria, . . 

Rebel e, . . 

9587 

Malta, . . 

Sal me, . . . 

16930 

. “ 

Kislos, . . 

10418 

Marseilles, 

Charge, . . 

9411 

Algiers, . . . 

Tarrie, . . . 

1219 

Milan, . . 

Moggi,. . . 

8444 

Amsterdam, . 

Mudde, . . 

6596 

Naples, . 

Temoli, . . 

3122 

44 

Sack, . . . 

4947 

Oporto, . . 

Alquiere, . . 

1051 

Antwerp, . . 

Viertel, . . 

4705 

Persia, 

Artaba. . . 

4013 

Azores, . . . 

Alquiere, . . 

731 

Poland, . . 

Zorzec, . . 

3120 

Berlin, . . . 

Scheffel, . . 

3180 

Riga, . . 

Loop, . . . 

3978 

Bremen, . „ . 

44 

4339 

Rome, . . 

Rubbio, . . 

16904 

Candia, . . . 

Charge, . . 

9288 

LI 

Quarti, . . 

4226 

Constantinople, 

Kislos,. . . 

2023 

Rotterdam, 

Sach, . . . 

6361 

Copenhagen, 

Toende, . . 

8489 

Russia, . . 

Clietwert, . 

12448 

Corsica, . . . 

Stajo, . . . 

6014 

Sardinia, 

Starelli, . . 

2988 

Florence, . . . 

Stari, . . 

1449 

Scotland, . 

Firlot, . . 

2197 

Geneva, . . . 

Coupes, . , 

4739 

Sicily, . 

Saline gros, . 

• 21014 

Genoa,.... 

Mina, . . 

7382 

LI 

generale, 

16886 

Greece, . . . 

Medimni,. . 

2390 

Smyrna, . 

Kislos,. . . 

2141 

Hamburg, . . 

Scheffel, . 

6426 

Spain, . . 

Catrize, . . 

41269 

Hanover, . . 

Malter, . . 

6868 

Sweden. . 

Tunna, . . 

8940 

Leghorn, . . . 

Stajo, . . 

1501 

Trieste, . . 

Stari, . . . 

4521 

44 

Sacco, . . . 

4503 

Tripoli, . 

Caffiri,. . . 

19780 

Lisbon, . . . 

Alquiere, . 

817 

Tunis, . . 

LL 

21855 

44 

• • • 

Fanega, . 

3268 

Venice. . 

Stajo, . . . 

4945 

Madeira, . . . 

Alquiere, . 

684 

Vienna, . . 

Metzen, . . 

3753 

Malaga, . . . 

Fanaga, . , 

3783 





English Measures of Capacity. 


The Imperial gallon measures 277-274 cubic inches, containing 10 lbs. 

Avoirdu- 

pois of distilled water, weighed in air, 

at the temperature of 62°, the barom- 

eter at 30 inches. 





For Grain. 

8 bushels = 1 quarter. 





1 quarter = 10-2694 cubic feet. 



Coal, or heaped measure. 

3 bushels 

= 1 sack. 




12 sacks 

= 1 chaldron 



Imperial bushel = 2218-192 cubic inches. 



*Heaped bushel, 19£ ins. diam., cone 6 ins. high = 2812-4872 cubic ins. 

1 chaldron 

= 58-658 cubic feet, and weighs 3136 pounds. 


1 chaldron (Newcastle) = 

5936 pounds. 








































Foreign Weights and Measures. 


43 


Foreign "Weights Compared with American. 


Places. 

Weights. 

Lbs. per 
100 avoir. 

Places. 

Weights. 

Lbs. per 
100 avoir. 

Aleppo,. . . 

Kottoli, . . 

20.46 

Hanover, . . 

Pound, . . 

93.20 


Oke, . . . 

35.80 

Japan, . . 
Leghorn, . . 

Catty, . . 

76.92 

Alexandria, . 

Rottoli, . . 

107. 

Pound, . . 

133.56 

Algiers, . . 

U 

84. 

Leipsic, . . 

“ (common) 

97.14 

Amsterdam, . 

Pound,. . . 

91.8 

Lyons, . . . 

“ (silk), 

98.81 

Antwerp, 

U 

96.75 

Madeira, . 


143.20 

Barcelona,. . 

a 

112.6 

Mocha, . . 

Maund, . . 

33.33 

Batavia, . . 

Catty, . . 

76.78 

Morea, . . 

Pound, . . 

90.79 

Bengal, . . . 

Seer, . . . 

53.57 

Naples, . . 

Rottoli, . . 

50.91 

Berlin, . . 

Pound, . . 

96.8 

Rome, . . 

Pound, . . 

133.69 

Bologna, . . 

(6 

125.3 

Rotterdam, . 

U 

91.80 

Bremen, . . 

u 

90.93 

Russia, . . 

U 

110.86 

Brunswick, . 

c< 

97.14 

Sicily, . . . 

H 

142.85 

Cairo, . . . 

Rottoli, . , . 

105. 

Smyrna, . 

Oke, . . . 

36.51 

Candia, . . . 

U 

85.9 

Sumatra, . . 

Catty, . 

35.56 

China, . . 

Catty, . . . 

75.45 

Sweden, . . 

Pound, . . 

106.67 

Constantinople 

Oke, . . . 

35.55 

u 

“ (miner’s), 

120.68 

Copenhagen, 

Pound, . . 

90.80 

Tangiers, . 
Tripoli, . . 

CC 

94.27 

Corsica, . . . 

U 

131.72 

Rottoli, . . 

89.28 

Cyprus, . . 

Rottoli, . . 

19.07 

Tunis, . . 

a 

90.09 

Damascus, . . 

6< 

25.28 

Venice, . . 

Pound (heavy) 

94.74 

Florence,. . 

Pound, . . . 

133.56 

U 

“ (light) 

150. 

Geneva, . . 

“ (heavy), 

82.35 

Vienna, . . 

U 

81. 

Genoa, . . 

tc U 

92.86 

Warsaw, 

U 

112.25 

Hamburgh, . 

u a 

93.63 





A Uniform System of Metrology much Needed. 

The preceding variety of tables of weights, measures and coins shows 
the great need of a uniform system of metrology throughout the world. 

The French are the first in adopting a uniform decimal system of 
metrology, and an International Decimal Association has been formed 
for the special purpose of advocating the introduction of the French 
system into other countries, which Association has now labored on that 
subject for some twenty years with hut little success. Only Belgium, 
Switzerland, Germany and Italy have adopted and enforced the French 
metrical system. It has been made legal in some other countries, but 
not enforced. 

The principal difficulties in the way appear to be prejudices and jeal¬ 
ousy. It must be admitted that the introduction of a new system of 
metrology causes some temporary inconveniences, but the objection is 
only temporary . Some few countries have decimated their old units in 
preference to adopting the French system. 

One difficulty of the decimal system is, that the base 10 does not admit 
of more than one binary division without fraction. See A New System 
of Arithmetic, page 44. 


























44 


Duodenal System. 


A NEW SYSTEM OF ARITHMETIC: WEIGHTS, 
MEASURES AND COINS. 

The discordance among nations in adopting a uniform system of weights, 
measures and coins is caused by the base of our decimal arithmetic not being 
well suited for that purpose. In the shop and market it is desired to divide 
the units into binary fractions, which cannot be accommodated by the base 
10 without cumberous decimal fractions, and the ordinary vulgar fractions 
are difficult to use in arithmetical calculations. 

The best system of metrology yet devised for decimal arithmetics is the 
French Metric System, which has been adopted by most civilized nations, 
but not by the most practical nations—namely, those who speak'the English 
language. The practical difficulty with the metric system is due to the arith¬ 
metic base 10 not admitting binary and trinary divisions without fractions, 
and that system is therefore inapplicable to the division of the circle and 
time, and can consequently not be used in navigation, geography and astron¬ 
omy. These incurable defects of the metric system can never make it stable, 
but its abolishment is a mere question of time. 

With the steady and unabated progress of sciences our descendants will 
not be contented with an incomplete system of metrology, but will devise a 
system that will accommodate the shop and market as well as geography and 
astronomy. The metric system is, however, far superior to the cumberous 
English metrology, and there is yet a chance for the English-speaking nations 
to devise and adopt a system of metrology that would be acceptable all over 
the world. Those who are thoroughly conversant with this subject seem to 
agree that the number 12 is the best suited as base for metrology, for the 
reason that it can be divided by 2, 3, 4 and 6 without leaving fractions, 
which is of great importance for the shop and market, and particularly so 
for mental calculations. 

A committee of the English Parliament, appointed for investigating the 
feasibility of adopting the metric system, reported that 12 as a base of me¬ 
trology would be much better for the shop and market than the metric 
system. 

A writer in the Edinburgh Review , January, 1807, advocated very strongly 
the adoption of 12 as the base not only for metrology, but also for arithmetic. 
This writer hoped that the wisdom of the English-speaking nations will 
soon come up to the level of this species of reform—namely, to adopt the 
number 12 as the base for all measurements. But such important improve¬ 
ments are not thought of in the ordinary course of human affairs. 

The Latin name for “twelve” is duodeni, and any system of metrology or 
arithmetic which is based on 12 is called the duodenal system,. 

A full description of duodenal systems of arithmetic and metrology will 
be found in Nystrom’s “Elements of Mechanics,” and also in Nystrom’s 
“ French Metric System.” 

The duodenal metrology with the present English units should be intro¬ 
duced first, and vrorked with decimal arithmetic until its utility is well under¬ 
stood, after which it would become very easy to introduce the duodenal 
arithmetic. 

The duodenal arithmetic is as much better than decimal arithmetic as 
decimal arithmetic is better than the Roman notation. 

The duodenal system has all the advantages and none of the disadvantages 
of the decimal system. 

The English-speaking nations are behind the other civilized nations in 
t-wo particular things—namely, in metrology and phonetic orthography. 
The removal of these two defects would save at least one half of the time, 
labor and expense now consumed in education. 

Our present systems of metrology and orthography are great burdens upon 
children to learn ; in fact, they can never learn them so as to be remembered, 
whilst the study of natural systems would be entertaining and never forgotten. 




Geometry. 


45 


GEOMETRY. 

DEFINITIONS. 

Demonstration is a course of reasoning by which a truth is established. It 
consists of, 

Thesis, the truth to be established, and, 

Hypothesis, the foundation for the demonstration. 

Axiom is that which is self-evident and requires no demonstration. 

Theorem is something to be proved by demonstration. 

Postulate is something to be done, but is self evident and requires no demon¬ 
stration. 

Problem is something proposed to be done, and requires demonstration. 
Proposition is either a Theorem or a Problem. 

Corolary is an obvious conseqence deduced from something that has gone 
before. 

Scolium is a remark on preceding propositions, commonly demonstrated fty 
algebraical formulae. 

Lemma is something premised for a following demonstration. 

Geometrical Quantities* 

Point is a position, but no magnitude. 

A Line is length, without breadth or thickness. 

A Straight Line is the shortest distance between two points. 

Curved line is a length which in every point changes its direction. 

Superficies, Surface, Area, is that which has length and breadth, but no 
thickness. 

Plane surface is a plane which coincides with a straight line in every direi 
tion. 

Curved surface is a plane, which coincides with a curved line. 

Solid has length, breadth and thickness. 

Circle* 

Circle, Cirumference, Periphery, is a curved line drawn on a plane surface, an/* 
bounded at a common distance from one point in the plane, (centre.) 

Radius is a line* drawn from the centre in a circle to the periphery. 

Diameter is a line drawn through the centre to the periphery, or the longesi 
line in a circle. 

Chord is any line extending its both ends to the periphery of a circle, and does 
not go through the centre. 

Arc is a part of a periphery. **■ 

Circle plane, is a plane surface bounded within a circumference. 

Sector is a part of a circle-plane bounded within an arc and two radii. 

Segment is a part of a circle plane bounded within a chord and an arc. 

Zone is a part of a circle included between two parallel chords. 

Lune is the space between the intersecting arcs of two eccentric circles. 

Oval is a round figure having one long and one short diameter at right angles 
to one another. 

Semicircle is a half circle. 

Quadrant is a quarter of a circle. 

Angies* 

Angle is the opening or inclination of two lines which meet in one point. 

If two radii being drawn from the extremities of a circle arc, to the centre; 
the arc, is a measure of the angle at the centre. 

Right angle is when the opening is a quarter of a circle. 

Acute angle is less than a right angle. 

Obtuse angle is greater than a right angle. 

* Line by itself means a straight line. 

1 












46 


Constructions. 




To construct an ellipse. 

With o as a centre, draw two concentric 
circles with diameters equal to the long and I 
short axes of the desired ellipse. Draw from j 
o any number of radii, A, B, &c. Draw the I 
line B b' parallel to n and b b' parallel to m, 
then b' is a point in the desired ellipse. 


2 . To draw an ellipse with a string. 

Having given the two axes, set .off from e 
half the great axis at a and b, which are the 
two focuses in the ellipse. Take an endless 
string as long as the three sides in the triangle 

a, b, c, fix two pins or nails in the focuses one 
in a, and one in b, lay the string round a, and 

b, stretch it with a pencil d, which then will 
describe the desired ellipse. 



To dr cow an ellipse by circle arcs. 


Divide the long axis into three equal parts, 
draw the two circles and where they intersect 
one another are the centres for the tangent 
arcs of the ellipse as shown by the figure. 


4 . To draw an ellipse by circle arcs. 

Given the two axes, set off the short axis 
from A to b , divide b B into three equal parts, 
set off two of these parts from o towards c 
and c which are the centres for the ends of 
the ellipse. Make equilateral triangles on c 
c, when e e will be the centres for the sides of 
the ellipse. If the long axis is more than 
tAvice the short one, this construction will not 
make a good ellipse. 



5 , To construct an ellipse. 

Given the two axes, set off half the long axis 
from c toff', which will be the two focuses in 
the ellipse. Divide the long axis into any num¬ 
ber of parts, say a to be a division point. Take 
A a as radius and / as centre and describe .a 
circle arc about b, take a B as radius and/ as 
centre describe another circle arc about b, then 
the intersection b is a point in the ellipse, and 
so the whole ellipse can be constructed. 



6 . 

To draw an ellipse that will tangent two 
parallel lines in A and B. 

Draw a semicircle on A B, draw ordinates 
in the circle at right angle to A B, the corre¬ 
sponding and equal ordinates for the ellipse 
to be drawn parallel to the lines, and thus the 
elliptic curve is obtained as shown by the 
figure. 













































Constructions. 


a 


47 



7« To construct a cycloid. 

The circumference C=3-14 D. Divide the 
rolling circle and base line C into a number 
of equal parts, draw through the division 
point the ordinates and abscissas, make a a? 
=1/ d, b b'= 2' e, c c'=3 /, then a’ b' and d are 
points in the cycloid. In the Epicycloid and 
Hypocycloid the abscissas are circles and the 
ordinates are radii to one common centre. 




8. Evolute of a circle . 

Given the pitch p, the angle v, and radius r. 
Divide the angle v into a number of equal 
parts, draw the radii and tangents for each 
part, divide the pitch p into an equal number 
of equal parts, then the first. tangent will be 
one part, second two parts, third three parts, 
&c., and so the Evolute is traced. 


9. To construct a spiral with compasses 
and four centres. 

Given the pitch of the spiral, construct a 
square about the centre, with the four sides 
together equal to the pitch. Prolong the 
sides in one direction as shown by the figure, 
the corners are the centres for each arc of the 
external angles. 



10. To construct a Parabola. 

Given the vertex A, axis x, and a point P. 
Draw A B at right angle to x, and B P parallel 
to x, divide A B and B P into an equal num¬ 
ber of equal parts. From the vertex A draw 
lines to the divisions on B P, from the divi¬ 
sions onAB draw the ordinates parallel to x, 
the corresponding intersections are points in 
the parabola. 



11. To construct a Parabola. 

Given the axis of ordinate B, and vertex A. 
Take A as a centre and describe a semicircle 
from B which gives the focus of the parabola at 
/. Draw any ordinate y at right angle to the 
abscissa A x, take a as radius and the focus/ 
as a centre, then intersect the ordinate y , by 
a circle-arc in P which will be a point in the 
parabola. In the same manner the whole 
Parabola is constructed. 


12 . 



To draw an arithmetic spiral. 

Given the pitch p and angle v, divide them 
into an equal number of equal parts say 6. 
make 01 = 01, 0 2=*02, 03=0 3, 04=04, 05=05, 
and 0 6=the pitch p ; then join the points 1 , 2, 
3, 4, 5, and 6, which will form the spiral re¬ 
quired. 



















































The Circle. 


<8 


THE CIRCLE. 

Notation of Letters. 

d = diameter of the circle, 
r = radius of the circle. 
p — periphery or circumfer¬ 
ence. 

a = area of a circle or part 
thereof. 

b = length of a circle-arc. 

All measures must be expressed by the same unit. 


c = chord of a segment, length 
of. 

h = height of a segment, 
s == side of a regular polygon. 
v — centre angle. 
w = polygon angle.. 


Formulas for the Circle. 


Periphery or Circum¬ 
ference. 

p = 7r d = 3.14d 
p = 2ir r = 6.28r. 
p = 2j/ t a = 

3 . 54^/07 

_2 a 4a 

r d 


Diameter and Radius. 

<* = £ = -£_. 

*• 3.14 

P p 

V ~ 2tt ~ 6 . 28 * 

d= 2 J-= 1 . 128 / 0 . 

M 7r 

r = A p = 0.564/^. 

\ 7T 


Area o/* the Circle. 

7T (T* 

a = —— = 0.785c? 2 . 
4 

a — 7r r' 2 = 3.14r 2 . 

_p 2 _ p 2 

“ ~ 4tt ~ 12.56 ’ 

_pjr_ p 

a — T~T* 


7T = 3.141 59 2 6 5 3589793238462643383279502884197169399 

360 


2tt=6.283185 

3tt=9.424778 

4tt=12.566370 

5tt=15.707963 

6:7=18.849556 

7:r=21.991148 

8tt=25.132741 

9^=28.274334 


*77=0.785398 

|tt=1.047197 

^tt=1.570796 

Itt^O.392699 

1tt=0.523599 

1^=0.261799 


1 

7r 

2 
7 r 

3 

77 

4 

77 

6 
7r 

|=2.546478 
^ =3.819718 


=0.318310 

=0.636619 

=0.954929 

=1.273239 

=1.909859 


=114.5915 

71 

tt 2 =9.869650 

^77=1.772453 




-=0.564189 


\/^=l.253314 

\ 2 




-=0.797884 

77 


Log. 77= 

0.49714987 













Longimetkt. 49 



4 
































50 


Longimetry. 


W_. 

1 y-iW/y 


19. 

i’ = r, w = w, 

tv + v = 180°, u> > t>. 



20. 

/b\ 


£ - 5 + C, + £' + C= 180°, 


/ 

B = D-C, A+B+ 0=180°, 

P/a: cVa 

A ' = A, B' = B'. 



21 . 


A + B + C = 180°, 
A'-A, B ' =11. 



22 . 


E+ C=A+D= 180°, 
D = B + c, 

22 = .4 + 21 . 


" ^ ' 

t- 1 

a 

O' 

ab 


23. 


(a + b) % “ a* + 2a6 + 


24. 


i"''' 

r U 

(0,-f 2 

A 

/ V ’ 


(a — &)* = a a — 2a5 + Z>\ 





























Longimetry 


51 












































52 


Longimetry. 


>N 

-.. 

f -r 

« 

• x 

\/ 

31. 

e = a* + i # , 
a* = c* - 6% 

A* = c* - a*. 

i 

K 


32. 

c® ■= a’ +A*~2M f 

* 

h = y/ a? — d*. 

j a* + V* — c* t 

2b 

1 \ \£ 

L,✓ V &\ 

33. 

c* = 0 * + A* + 2AJ, 

A a = V a® — <i®, 

c* - a 3 - A® 
d “ 26 

/"Z^% 

! / i V 

// l J \ 

<(f, > 

34. 

a : b = h : c, 

ac ad 
h = -? = —* 

A C 

C * cA 

A a 

JP/V 

A - A 

35. 

a : c = d : (b — d), 

, ab 

d =-* 

c + a 

V = V. 

© 

36. 

a : c = b : d 9 

ad = be . 

-... .. i 































Longimetry. 


53 
































54 


Longimetky. 



J M O ) | 


To find the length of a Spiral. 

, Trr* l r 

l = „m= T , n=- = r 

P=«l’-i. f-PifcA. 

1 n 


\(vv x yy) 


44. To find the length of a Spiral. 

1 = it n (ft +r), 

i- ?(&-<*)■ 


rx 

45. Periphery of an Ellipse. 

vj 

jv 

= 2 \/ i) a + l'4674d\ 

5/ 

/ 

4^, —' 

> 

\ 

46. 

To construct a screw Helix. 


2 

i""" 



A ; 

1 

V ^ 

/'X. 1 

X >> 

w 

n / 

< 

V 

4^* To square a Circumference. 

R = 0-555355 d = 1-1107 r - 0-7071 S. 

S = 0-785398 = 1-57079 r = 1-4142 R. 

d = 1-27322 S = 1-79740 R = 2 r. 

si 

V 

.. BiS 

A 

l| rf 

48* To square a Cirdeplane. 

R = 0-626657 d = 1-253314 r = 0-7071 S. 

S - 0-886226 = 1-77245 r -1-4142 ft 

^ = 1-12838 S = 1-5367 ft = 2 r. 


































Polyhedrons. 


65 



Tetrahedron. 

r = 0-20413 g. 

R — 0-61237 s. 
a = 1-73205 s’, 
c = 0-11785 s*. 


Hexahedron. 

r = 0-50000 s. 
R = 0-86602 s. 
a = 6-00000 g«. 
c = 1-00000 s*. 


J 

51. Octahedron. 

r = 0-40721 s. 

R = 0-70710 s. 
a = 3-46410 s*. 
c = 0-47140 s* 


52. Dodecahedron. 

r = 1-11350 s. 

R = 1-40122 s. 
a = 20-6457 s*. 
c = 7-66312 s». 

0mA 

53. Icosahedron. 

r = 0-7558 s. 

R = 0-9510 s. 
a = 8-66025 s\ 
c = 2-18169 s*. 

r = Radius of an inscribed Sphere. 

R — Radius of circumscribed Sphere, 
a = Area of the Polyhedrons, 
c = Cubic contents of the Polyhedrons, 
s = Side or edge of the Polyhedrons. 








































<56 


Planemetry. 



s > 

■ 

1 

A 

l 

54. Square, 

a = s a = 45*. 

a = 0-5 d 2 

< a 

55. Rectangle, 

a = a 5, 

a = 5 V - 5». 


56. Triangle, 

bh ,z x 

a = -- = 45 A, 

“=W--(" + 2 t c t* 


h 

* 

57. Triangle, 

a ~ 4 5 5, 

e a 

- - 

58. Quadrangle. 

a = 45(a + 5). 

Jll 

59. Quadrangle. 

a = 4(fl [A + A'] + 5 A' + e h). 




























PLANEMETKr. 


57 



60. Circle Plane. 

w 

a = tt r* = 0-785 d*, 

a = — = 0.0796 P a . 


61. Circle Ring. 

a = 7Z(iP — r a ) = + r)(R—r) t 

a = 0-785(2P-<P). 


62. Sector . 

a = Pr, 

Ttr 1 v » 

a “ 360 “114-5* 

-c. 

63. Segment. 

a = i-[5 r — c (r — A)], 

n r* v c , i\ 

a “ 360 +2^ ^ 

JS 

64. Quadrant. 

a = 0-785 r* = 0-3927 c a . 

j- 9 / j 

/ Ji 

65. 

a =0-215 r* = 0*1075 e\ 































58 


Planemetry. 




Ellipse. 

a = rtiJr= 0-785 D d. 

Parabola. 
a = f b h = £ b*, 
a = f kV p h. 
Irregular Figure. 

a =£(£ + // +A"). 



Ellipsoid. 

a = 8-88 r v/ - RHr’, 
a = 2-22d VirTd\ 


Cylinder. 


a = 2 Ji r h — it d h, 
j_ a _ a 

~ 2nr 7t d ‘ 



71.Tlie road to Extremity of Space* 

1st. Draw a Circle and inscribe a Square, and in 

that Square a Circle, &c,, &c., &c. The last 

figure that can be drawn, is one extremity of Space 
Required if the last one is a Circle or a Square? 

2nd. Draw a Circle and circumscribe a Square, 

and around that Square a Circle, &c., &c., &c. 

The last one that can be circumscribed is the other 
extremity of space. Required if the last figure is 
a Circle or a Square ? 




























































Bur-face of Solids. 


59 



74. Sphere Sector . 















































60 


Stereometry. 



78. Sphere. 

c = —J~"= 4-189 r*, 

c = 0-523 <i*. 

0 


79. Forus. 

c = 2 5 T 2 7? r 2 = 19*74 72 r% 


c= 2-463 D <iV 

^ggllg 

80. Sphere Sector. 

c = f TCr* h = 2-0944 r 3 A, 


c = f r 3 (r + \/y a - i c a ). 


81. Zone. 

c ^ fif A 3 (r — £ 




82. Cone. 

c-^g A = 1-047r 3 A, 

c = 0-2618 d a A. 


83. Conic Frustum. 

#f»s 

C = s ^ ^(R 3 + R r + r 2 )* 

c = T v^(z> a + R <* + <**) 



























Stereometry. 


61 



84. Cylinder. 

C = n r 3 A - 0.785 d 3 A, 

c-£±- 00796 p’ 

4 7Z ^ 

f R-> 

85. Ellipsoid. 


C = 0-424 tt" R r 3 = 4-1847 12 r*. 

J) 

c = 0 053 n % I> d* = 0-5231 D d 3 

^ j 

86. Paraboloid. 

< T * 

c = £:tfr 3 A = 1-5707 r 3 A. 


87. Pyramid. 

c = iaA, 


n s A / . s 3 

e== T _ \/ r r* 

^ j 

V/ Y T 

GS 

88. Pyramidic Frustum. 

c = -^(A + a + N /ia). 


89. Frustum. 

hs, .» 

4T» —* , - / n t M 

^ Hi a 

c = 




























62 


Stereometry. 



90. Cash, 

c = 1-0453 i(0-4 X>’+ 0-2 D d + 0-15d*), 

Gallon - + 2 Dd + 1-5 d*). 



91. Cylinder Sections. 

C = Tt r*(l + /' — f r), 
c = Tt r\l + V) — 2*1 r*. 



92. 


Circular Spindle. 


c = ttQ c 3 — 0-2 c?[c+§ V c a +</ 2 ] v^M-c 5 ) 


Example 1. Fig. 56. The base of a Triangle is 5 = 8 feet, 3 inches, and the 
height, h = 5 feet, 6 inches. What is the area a = ? 


b h 8-25 X 5-5 


= 22-6875 square feet. 


2 2 

Example 2. Fig. 62. A Circle Sector having an angle v = 39° and the radius 
r = 67f inches. What is the area of the sector a = ? 

Tlr^v 3-J4X 67-75* X 39° 


a = 


360 


360 


= 1562T square feet. 


Example 3. Fig. 75. A Spherical Zone having its diameter c = 18| inches 
and height h = 7£ inches. What is the convex surface of the Zone l 


a =— ^c*-f ^18-5 9 + 7’75*^ = 315-96 square inches. 

Example 4. Fig. 52. Require the radius R of a Sphere that will circumscribe 
a Dodecahedron with the side s = 9 inches. 

R = 1-36428 X 9 = 12-27852 inches, the answer. 

Example 5. Fig. 89. A Frustrum of a Cone having its bottom diameter D = 13 
inches, the top diameter d — 5£ inches, and the height h — 2b inches. What is 
the cubic contents c = ? 

c = T V + Dd + 0-2618 X 25 (l3» + 13 X 5'25 + 5-25*)= 20995 

cubic inches. 

Example 6. Fig. 90. A Cask having its bung diameter D — 36 inches, head 
diameter d = 28 inches, and length l — 56 inches, (inside measurement) how 
many gallons of liquid can be contained in the cask ? (The gallon = 231 cub. it_.) 

Gallon = -~ q (4 X 36* + 2 X 36 X 28 + D5 X 28*)= 214 gallons. 
























































G-eojietry.—Table of Polygons. 


63 


Example 7. Fig. 14. Require the length of the circle-arc b, when the angle 
v — 42°, and the radius r — 4 feet, 3 inches ? 

_wr r_ 3-14X4 -25X42 _ 
b ~ 180 “ 180 “ 3 113 feet * 

Example 8. Fig. 16. Require the radius of a circle-arc, whose chord is 9 feet, 
4 inches, and height, h — 1 foot, 8 inches ? 


r = 


c 2 +4A 2 9-332-F4Xl-66 a 98-0711 


= 7-384 feet. 


8 h 8XP66 13-28 

Example 9. Fig. 32. The three sides in a triangle being, a = 6*42, 6 


and c == 8-66 feet. How high is the triangle over the base 6? 


7*75, 


d = 


a 2+&2_c2 6-422+7-75 2 — 8-662 


26 


2X8-66 


= 1-5175 feet, 


the height h — +a 2 — dfi = y/6.42 2 —1-5175 2 = 6-24 feet, the answer. 

Example 10. Fig. 41. The radius of a walking beam is, r — 8*36 feet, the stroke 
S = 5-5 feet. How much is the vibration V — ? 


Vibration, 


Y—r — r* — $1. = 8-36 — 8'36 2 

21 

=0-471 feet = 5'65 inches = 5 - —, the answer. 

32 ’ 


TABLE OP POLYGONS. 


Number 
of sides 
in the 
Polygon. 


Tngon. 

Tetragon. 

Pentagon. 

Hexagon. 

Heptagon. 

Octagon. 

Nonagon. 

Decagon. 

Undecagon. 

Dodecagon. 


Centre 
Angle tc. 



3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

14 

15 

16 
18 
20 
24 


120 ° 

90° 

72° 

60° 
51°43' 
45° 

40° 

36° 

32°13' 

30° 

25°43' 

24° 

22°30' 

20 ° 

18° 

15° 


Polygon 
Angle v. 



60° 

90° 
108° 
120 ° 
128°17' 
135° 
140° 
144° 
147°47' 
150° 
154°17' 
156° 
157°30' 
160° 
162° 
165° 


Side 

= k R. 



1*732 

1-4142 

1-1755 

1-0000 

0-8677 

0-765.3 

0-6840 

0-6180 

0-5634 

0-5176 

0-4450 

0-4158 

0-3900 

0-3472 

0-3130 

0-2610 


Are* 
— h S®. 



0-4330 

1-0000 

1- 7205 

2- 5980 

3- 6339 

4- 8284 
6-1820 
7-6942 
9-3656 

11-196 

15-334 

17-642 

20-128 

25-534 

40-634 

45-593 


Apotem 

= HR. 



0-5000 

0-7071 

0-8090 

0-8660 

0-9009 

0-9238 

0-9396 

0-9510 

0-9595 

0-9659 

0-9762 

0-9781 

0-9807 

0-9848 

0-9877 

0-9914 


Side 

-hr 



3-4641 

2-0000 

1-4536 

1-1547 

0-9631 

0-8284 

0-7279 

0-6498 

0-5872 

0-5359 

0-4562 

0-^250 

0-4068 

0-3526 

0-3166 

0-2632 


Area 

= A r a . 



5-1961 

4-0000 

3-6327 

3-4640 

3-3710 

3-3130 

2750 

2490 

2290 

2152 

1935 

3-1882 

3-1824 

3-1737 

3-1676 

3-1596 


Explanation of tlie Tal»le for Polygons. 

The number of sides in the polygon is noted in the first column. 

7<; = tabular coefficient, to be multiplied as noted on the top of the columns. 
Example 1. How long is the side of an inscribed Pentagon, when the radius 
of the circle is 3 feet, and 4 inches ? (4 inches == 0-333 feet.) 

3-333X1-1755 = 3-9179 feet, the answer. 

Example 2. What is the area of a Heptagon when one of its sides is 13-75 in ches 
13 - 75 a X3'6339—687‘02 square inches. 



























































64 


CIRCUMFERENCE AND AREA OF CIRCLES. 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Area. 

Diam- 



Diam- 



Diam- 


Infill 

eter. 

U 

llP 

eter. 


Ijjp 

eter. 

u 

IIP 

1 

3-1416 

0-7854 

51 

160-22 

2042-8 

101 

317-30 

8011-9 

2 

6-2832 

3-1416 

52 

163-36 

2123.7 

102 

320-44 

8171-3 

3 

9-4248 

7-0686 

53 

166-50 

2206.2 

103 

323-58 

8332-3 

4 

12-566 

12-5664 

54 

169-65 

2290-2 

104 

326-73 

8494-9 

5 

15-708 

19-6350 

55 

172-79 

2375-8 

105 

329-87 

8659-0 

6 

18-850 

28-2743 

56 

175-93 

2463-0 

106 

333-01 

8824-7 

7 

21-991 

38-4845 

57 

179-07 

2551-8 

107 

336-15 

8992-0 

8 

25-133 

50-2655 

58 

182-21 

2642-1 

108 

339-29 

9160-9 

9 

28-274 

63-6173 

59 

185-35 

2734-0 

109 

342-43 

9331-3 

10 

31-416 

78-54 

60 

188-50 

2827-4 

110 

345-58 

9503-3 

11 

34-558 

95-03 

61 

191-64 

2922-5 

111 

348-72 

9676-9 

12 

37-699 

113-10 

62 

194-78 

3019-1 

112 

351-86 

9852.0 

13 

40-841 

132-73 

63 

197-92 

3117-2 

113 

355.00 

10028-8 

14 

43-982 

153-94 

64 

201-06 

3217-0 

114 

358-14 

10207-0 

15 

47-124 

176-71 

65 

204-20 

3318-3 

115 

361-28 

10386-9 

16 

50-265 

201-06 

66 

207-35 

3421-2 

116 

364-42 

10568.3 

17 

53-407 

226-98 

67 

210-49 

3525-7 

117 

367-57 

10751-3 

18 

56-549 

254-47 

68 

213-63 

3631-7 

118 

370-71 

10935-9 

19 

59-690 

283-53 

69 

216-77 

3739-3 

119 

373-85 

11122-0 

20 

62-832 

314-16 

70 

219-91 

3848-5 

120 

376-99 

11310 

21 

65-973 

346-36 

71 

223-05 

3959-2 

121 

380.13 

11499 

22 

69-115 

380-13 

72 

226-19 

4071.5 

122 

383-27 

11690 

23 

72-257 

415-48 

73 

229-34 

4185-4 

123 

386-42 

11882 

24 

75-398 

452-39 

74 

232-48 

4300-8 

124 

389-56 

12076 

25 

78-540 

490-87 

75 

235.62 

4417-9 

125 

392-70 

12272 

26 

81*631 

530-93 

76 

238-76 

4536-5 

126 

395-84 

12469 

27 

84-823 

572-56 

77 

241-90 

4656-6 

127 

398-98 

12668 

28 

87-965 

615-75 

78 

245-04 

4778-4 

128 

402-12 

12868 

29 

91-106 

660-52 

79 

248-19 

4901-7 

129 

405-27 

13070 

30 

94-248 

706-86 

80 

251-33 

5026-6 

130 

408-41 

13273 

31 

97-389 

754-77 

81 

254-47 

5153-0 

131 

411-55 

13478 

32 

100-53 

804-25 

82 

257-61 

5281-0 

132 

414-69 

13685 

33 

103-67 

855-30 

83 

260-75 

5410-6 

133 

417-83 

13893 

34 

106-81 

907-92 

84 

263-89 

5541-8 

134 

420-97 

14103 

35 

109-96 

962-11 

85 

267.04 

5674-5 

135 

424-12 

14314 

36 

113-10 

1017-88 

86 

270-18 

5808-8 

136 

427.26 

14527 

37 

116-24 

1075-21 

87 

273-32 

5944-7 

137 

430-40 

14741 

38 

119-38 

1134-11 

88 

276-46 

6082-1 

138 

433-54 

14957 

39 

122-52 

1194-59 

89 

279-60 

6221-1 

139 

436-68 

15175 

40 

125-66 

1256-63 

90 

282-74 

6361-7 

140 

439-82 

15394 

41 

128-81 

1320-25 

91 

285-88 

6503-9 

141 

442-96 

15615 

42 

131-95 

1385-44 

92 

289-03 

6647-6 

142 

446*11 

15837 

43 

135-09 

1452-20 

93 

292-17 

6792-9 

143 

449-25 

16061 

44 

138-23 

1520-52 

94 

295-31 

6939-8 

144 

452-39 

16286 

45 

141-37 

1590-43 

95 

298-45 

7088-2 

145 

455-53 

16513 

46 

144-51 

1661-90 

96 

301-59 

7238-2 

146 

458-67 

16742 

47 

147-65 

1734-94 

97 

304-73 

7389-8 

147 

461-81 

16972 

48 

150-80 

1809-55 

98 

307-88 

7543-0 

148 

464-96 

17203 

49 

153.94 

1885-74 

99 

311-02 

7697-7 

149 

468-10 

17437 

50 

157-08 

1963-5 

100 

314-16 

7854-0 

150 

471-24 

17671 



















Circumference and Area of Circles. 


65 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Area. 

Diam¬ 

( > 


Diam¬ 


Jgg|, 

Diam¬ 



eter. 


iJjP 

eter. 

u 

IIP 

eter. 

o 

IIP 

151 

474*38 

17908 

201 

631-46 

31731 

251 

788-54 

49481 

152 

477-52 

18146 

202 

634-60 

32047 

252 

791-68 

49876 

153 

480-66 

18385 

203 

637-74 

32365 

253 

794-82 

50273 

154 

483-81 

18627 

204 

640-89 

32685 

254 

797-96 

50671 

155 

486-95 

18869 

205 

644-03- 

33006 

255 

801-11 

51071 

156 

490-09 

19113 

206 

647-17 

33329 

256 

804-25 

51472 

157 

493-23 

19359 

207 

650-31 

33654 

257 

807-39 

51875 

158 

496-37 

19607 

208 

653-45 

33979 

258 

810-53 

52279 

159 

499-51 

19856 

209 

656-59 

34307 

259 

813-67 

52685 

160 

502-65 

20106 

210 

659-73 

34636 

260 

816-81 

53093 ' 

161 

505-80 

20358 

211 

662-88 

34967 

261 

819-96 

53502 

162 

508-94 

20612 

212 

666-02 

35299 

262 

823-10 

53913 

163 

512-08 

20867 

213 

669-16 

35633 

263 

826-24 

54325 

164 

515-22 

21124 

214 

672-30 

35968 

264 

829-38 

54739 

165 

518-36 

21382 

215 

675-44 

36305 

265 

832-52 

55155 

166 

521-50 

21642 

216 

678-58 

36644 

266 

835-66 

55572 

167 

524-65 

21904 

217 

681-73 

36984 

267 

838-81 

55990 

168 

527-79 

22167 

218 

684-87 

37325 

268 

841-95 

56410 

169 

530-93 

22432 

219 

688-01 

37668 

269 

845-09 

56832 

170 

534-07 

22698 

220 

691-15 

38013 

270 

848-23 

57256 

171 

537-21 

22966 

221 

694-29 

38360 

271 

851-37 

57680 

172 

540-35 

23235 

222 

697-43 

38708 

272 

854-51 

58107 

173 

543-50 

23506 

223 

700-58 

39057 

273 

857*66 

58535 

174 

546-64 

23779 

224 

703-72 

39408 

274 

860-80 

58965 

175 

549-7S 

24053 

225 

706-86 

39761 

275 

863-94 

59396 

176 

552-92 

24328 

226 

710-00 

40115 

276 

867-08 

59828 

177 

556-06 

24606 

227 

713-14 

40471 

277 

870-22 

60263 

178 

559-20 

24885 

228 

716-28 

40828 

278 

873-36 

60699 

179 

562-35 

25165 

229 

719-42 

41187 

279 

876-50 

61136 

180 

565-49 

25447 

230 

722-57 

41548 

280 

879-65 

61575 

181 

568-63 

25730 

231 

725-71 

41910 

281 

882-79 

62016 

182 

571-77 

26016 

232 

728-85 

42273 

282 

885-93 

62458 

183 

574-91 

26302 

233 

731-99 

42638 

283 

889-07 

62902 

184 

578-05 

26590 

234 

735-13 

43005 

284 

892-21 

63347 

185 

581-19 

26880 

235 

738-27 

43374 

285 

895-35 

63794 

186 

584-34 

27172 

236 

741-42 

43744 

286 

898-50 

64242 

187 

587-48 

27465 

237 

744-56 

44115 

287 

901-64 

64692 

188 

590-62 

27759 

238 

747-70 

44488 

288 

904-78 

65144 

189 

593-76 

28055 

239 

750-84 

44863 

289 

907-92 

65597 

190 

596-90 

28353 

240 

753-98 

45239 

290 

911-06 

66052 

191 

600-04 

28652 

241 

757-12 

45617 

291 

914-20 

66508 

192 

603-19 

28953 

242 

760-27 

45996 

292 

917-35 

66966 

193 

606-33 

29255 

243 

763-41 

46377 

293 

920-49 

67426 

194 

609-47 

29559 

244 

766-55 

46759 

294 

923-63 

67887 

195 

612-61 

29865 

245 

769-69 

47144 

295 

926-77 

68349 

196 

615-75 

30172 

2^6 

772-83 

47529 

296 

929-91 

68813 

197 

618-89 

30481 

247 

775-97 

47916 

297 

933-05 

69279 

198 

622-04 

30791 

248 

779-12 

48305 

298 

936-19 

69747 

199 

625-18 

31103 

249 

782-26 

48695 

299 

939-34 

70215 

200 

628-32 

31416 

250 

785-40 

49087 

300 

942-48 

70686 


5 














66 Circumference and Are^ of Circles. 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Area. 

Diam- 



Diam- 



Diam- 



eter. 

u 

ISP 

eter. 

w 

flip 

eter. 

Kj 


301 

945*62 

71158 

351 

1102*70 

96762 

401 

1259*78 

126293 

302 

948*76 

71631 

352 

1105*84 

97314 

402 

1262*92 

126923 

303 

951*90 

72107 

353 

1108*98 

97868 

403 

1266*06 

127556 

304 

955*04 

72583 

354 

1112*12 

98423 

404 

1269*20 

128190 

305 

958*19 

73062 

355 

1115*27 

98980 

405 

1272*35 

128825 

306 

961*33 

73542 

356 

1118*41 

99538 

406 

1275*49 

129462 

307 

964*47 

74023 

357 

1121*55 

100098 

407 

1278*63 

130100 

308 

967*61 

74506 

358 

1124*69 

100660 

408 

1281*77 

130741 

309 

970*75 

74991 

359 

1127*83 

101223 

409 

1284*91 

131382 

310 

973*89 

75477 

360 

1130*97 

101788 

410- 

1288*05 

132025 

311 

977*04 

75964 

361 

1134*11 

102354 

411 

1291*19 

132670 

312 

980*18 

76454 

362 

1137*26 

102922 

412 

1294*34 

133317 

313 

983*32 

76945 

363 

1140*40 

103491 

413 

1297*48 

1339-65 

314 

986*46 

77437 

364 

1143*54 

104062 

414 

1300*62 

134614 

315 

989*60 

77931 

365 

1146*68 

104635 

415 

1303*76 

135265 

316 

992*74 

78427 

366 

1149*82 

105209 

416 

1306*90 

135918 

317 

995-S8 

78924 

367 

1152*96 

105785 

417 

1310*04 

136572 

318 

999*03 

79423 

368 

1156*11 

106362 

418 

1313*19 

137228 

319 

1002*17 

79923 

369 

1159*25 

106941 

419 

1316*33 

137885 

320 

1005*31 

80425 

370 

1162*39 

107521 

420 

1319*47 

138544 

321 

1008*45 

80928 

371 

1165*53 

108103 

421 

1322*61 

139205 

322 

1011*59 

81433 

372 

1168*67 

108687 

422 

1325*75 

139867 

323 

1014*73 

81940 

373 

1171*81 

109272 

423 

1328*89 

140531 

324 

1017*88 

82448 

374 

1174*96 

109858 

424 

1332*04 

141196 

325 

1021*02 

82958 

375 

1178*10 

110447 

425 

1335*18 

141863 

326 

1024*16 

83469 

376 

1181*24 

111036 

426 

1338*32 

142531 

327 

1027*30 

S3982 

377 

1184*38 

111628 

427 

1341*46 

143201 

328 

1030*44 

84496 

378 

1187*52 

112221 

428 

1344*60 

143872 

329 

1033*58 

85012 

379 

1190*66 

112815 

429 

1347*74 

144545 

330 

1036*73 

85530 

380 

1193*81 

113411 

430 

1350*88 

145220 

331 

1039*87 

86049 

381 

1196*95 

114009 

431 

1354*03 

145896 

332 

1043*01 

86570 

382 

1200*09 

114608 

432 

1357*17 

146574 

333 

1046*15 

87092 

383 

1203*23 

115209 

433 

1360*31 

147254 

334 

1049*29 

87616 

384 

1206*37 

115812 

434 

1363*45 

J47934 

335 

1052*43 

88141 

385 

1209*51 

116416 

435 

1366*59 

148617 

336 

1055*58 

88668 

386 

1212*65 

117021 

436 

1369*73 

149301 

337 

1058*72 

89197 

387 

1215*80 

117628 

437 

1372*88 

149987 

338 

1061*86 

89727 

388 

1218*94 

118237 

438 

1376*02 

150674 

339 

1065*00 

90259 

389 

1222*08 

118847 

439 

1379*16 

151363 

340 

1068-14 

90792 

390 

1225*22 

119459 

440 

1382*30 

152053 

341 

1071*28 

91327 

391 

1228*36 

120072 

441 

1385*44 

152745 

342 

1074*42 

91863 

392 

1231*50 

120687 

442 

1388*58 

153439 

343 

1077*57 

92401 

393 

1234*65 

121304 

443 

1391*73 

154134 

344 

1080*71 

92941 

394 

1237*79 

121922 

444 

1394*87 

154830 

345 

1083*85 

93482 

395 

1240-93 

122542. 

445 

1398-01 

155528 

346 

1086*99 

94025 

396 

1244*07 

123163 

446 

1401*15 

156228 

347 

1090*13 

94569 

397 

1247*21 

123786 

447 

1404*29 

156930 

348 

1093*27 

95115 

398 

1250*35 

124410 

448 

1407*43 

157633 

349 

1096*42 

95662 

399 

1253*50 

125036 

449 

1410*58 

158337 

350 

1099*56 

96211 

400 

1256*64 

125664 

450 

1413*72 

159043 























Circumference and Area of Circles. 


67 



Circum. 

Area. 

Diam¬ 



eter. 

o 

IIP 

451 

1416-86 

159751 

452 

1420-00 

160460 

453 

1423-14 

161171 

454 

1426-28 

161883 

455 

1429-42 

162597 

456 

1432-57 

163313 

457 

1435-71 

164030 

458 

1438-85 

164748 

459 

1441-99 

165468 

460 

1445-13 

166190 

461 

1448-27 

166914 

462 

1451-42 

167639 

463 

1454-56 

168365 

464 

1457-70 

169093 

465 

1460-84 

169823 

466 

1463-98 

170554 

467 

1467-12 

171287 

468 

1470-27 

172021 

469 

1473-41 

172757 

470 

1476-55 

173494 

471 

1479-69 

174234 

472 

1482-83 

174974 

473 

1485-97 

175716 

474 

1489-11 

176460 

475 

1492-26 

177205 

476 

1495-40 

177952 

477 

1498-54 

178701 

478 

1501-68 

179451 

479 

1504-82 

180203 

480 

1507-96 

180956 

481 

1511-11 

181711 

482 

1514-25 

182467 

483 

1517-39 

183225 

484 

1520-53 

183984 

485 

1523-67 

184745 

486 

1526-81 

185508 

487 

1529-96 

186272 

488 

1533-10 

187038 

489 

1536-24 

187805 

490 

1539-38 

188574 

491 

1542-52 

189345 

492 

1545-66 

190117 

493 

1548-81 

190890 

494 

1551-95 

191665 

495 

1555-09 

192442 

496 

1558-23 

193221 

497 

1561-37 

194000 

498 

1564-51 

194782 

499 

1567-65 

195565 

500 

1570-80 

196350 



Circum. 

Area. 

Diam¬ 

eter. 

n 

n§ 


501 

1573-94 

197136 

502 

1577-08 

197923 

503 

1580-22 

198713 

504 

1583-36 

199504 

505 

1586-50 

200296 

506 

1589-65 

201090 

507 

1592-79 

201886 

508 

1595-93 

202683 

509 

1599-07 

203482 

510 

1602-21 

204282 

511 

1605-35 

205084 

512 

1608-50 

205887 

513 

1611-64 

206692 

514 

1614-78 

207499 

515 

1617-92 

208307 

516 

1621-06 

209117 

517 

1624-20 

209928 

518 

1627-35 

210741 

519 

1630-49 

211556 

520 

1633-63 

212372 

521 

1636-77 

213189 

522 

1639-91 

214008 

523 

1643-05 

214829 

524 

1646-20 

215651 

525 

1649-34 

216475 

526 

1652-48 

217301 

527 

1655-62 

218128 

528 

1658-76 

218956 

529 

1661-90 

219787 

530 

1665-04 

220618 

531 

1668-19 

221452 

532 

1671-33 

222287 

533 

1674-47 

223123 

534 

1677-61 

223961 

535 

1680-75 

224801 

536 

1683-89 

225642 

537 

1687-04 

226484 

538 

1690-18 

227329 

539 

1693-32 

228175 

540 

1696-46 

229022 

541 

1699*60 

229871 

542 

1702-74 

230722 

543 

1705-88 

231574 

544 

1709-03 

232428 

545 

1712-17 

233283 

546 

1715-31 

234140 

547 

1718-45 

234998 

518 

1721-59 

235858 

549 

1724-73 

236720 

550 

1727-88 

237583 



Circum. 

Area. 

Diam- 



eter. 

o 

lip 

551 

1731-02 

238448 

552 

1734-16 

239314 

553 

1737*30 

240182 

554 

1740-44 

241051 

555 

1743-58 

241922 

556 

1746-73 

242795 

557 

1749-87 

243669 

558 

1753-01 

244545 

559 

1756-15 

245422 

560 

1759-29 

246301 

561 

1762-43 

247181 

562 

1765-58 

248063 

563 

1768-72 

248947 

564 

1771-86 

249832 

565 

1775-00 

250719 

566 

1778-14 

251607 

567 

1781-28 

252497 

568 

1784-42 

253388 

569 

1787-57 

254281 

570 

1790-71 

255176 

571 

1793-85 

256072 

572 

1796-99 

256970 

573 

1800-13 

257869 

574 

1803-27 

258770 

575 

1806-42 

259672 

576 

1809-56 

260576 

577 

1812-70 

261482 

578 

1815-84 

262389 

579 

1818-98 

263298 

580 

1822-12 

264208 

581 

1825-27 

265120 

582 

1828-41 

266033 

583 

1831-55 

266948 

584 

1834-69 

267865 

585 

1837-83 

268783 

586 

1840-97 

269702 

587 

1844-11 

270624 

588 

1847-26 

271547 

589 

1850-40 

272471 

590 

.1853-54 

273397 

591 

1856-68 

274-325 

592 

1859-82 

275254 

593 

1862-96 

276184 

594 

1866-11 

277117 

595 

1869-25 

278051 

596 

1872-39 

278986 

597 

1875-53 

279923 

598 

1878-67 

280862 

599 

1881-81 

281802 

600 

1884-96 

282743 
















63 


Circumference and Area of Circles. 


t , 

Circum. 

Area. 

1 

Circum. 

Diam¬ 

eter. 

6 

0 

Diam¬ 

eter. 

o 

601 

1888-10 

283687 

651 

2045-18 

602 

1891-24 

284631 

652 

2048-32 

603 

1894-38 

285578 

653 

2051-46 

604 

1897-52 

286526 

654 

2054-60 

605 

1900-66 

287475 

655 

2057-74 

606 

1903-81 

288426 

656 

2060-88 

607 

1906-95 

289379 

657 

2064-03 

608 

1910-09 

290333 

658 

2067-17 

609 

1913-23 

291289 

659 

2070-31 

610 

1916-37 

292247 

660 

2073-45 

611 

1919-51 

293206 

661 

2076-59 

612 

1922-65 

294166 

662 

2079-73 

613 

1925-80 

295128 

663 

2082-88 

614 

1928-94 

296092 

664 

2086-02 

615 

1932-08 

297057 

665 

2089-16 

616 

1935-22 

298024 

666 

2092-30 

617 

1938-36 

298992 

667 

2095-44 

618 

1941-50 

299962 

668 

2098-58 

619 

1944-65 

300934 

669 

2101-73 

620 

1947-79 

301907 

670 

2104-87 

621 

1950-93 

302882 

671 

2108-01 

622 

1954-07 

303858 

672 

2111-15 

623 

1957-21 

304836 

673 

2114-29 

624 

1960-35 

305815 

674 

2117-43 

625 

1963-50 

306796 

675 

2120-58 

626 

1966-64 

307779 

676 

2123-72 

627 

1969-78 

308763 

677 

2126-86 

628 

1972-92 

309748 

678 

2130-00 

629 

1976-06 

310736 

679 

2133-14 

630 

1979-20 

311725 

680 

2136-28 

631 

1982-35 

312715 

681 

2139-42 

632 

1985-49 

313707 

682 

2142-57 

633 

1988-63 

314700 

683 

2145-71 

634 

1991-77 

315696 

684 

2148-85 

635 

1994-91 

316692 

685 

2151-99 

636 

1998-05 

317690 

686 

2155-13 

637 

2001-19 

318690 

687 

2158-27 

638 

2004-34 

319692 

688 

2161-42 

639 

2007-48 

320695 

689 

2164-56 

640 

2010-62 

321699 

690 

2167-70 

641 

2013-67 

322705 

691 

2170-84 

642 

2016-90 

323713 

692 

2173-98 

643 

2020-04 

324722 

693 

2177-12 

644 

2023*19 

325733 

694 

2180-27 

645 

2026-33 

326745 

695 

2183-41 

646 

2029-47 

327759 

696 

2186-55 

647 

2032-61 

328775 

697 

2189-69 

648 

2035-75 

329792 

698 

2192-83 

649 

2038-89 

330810 

699 

2195-97 

650 

2042-04 

331831 

700 

2199-11 


Area. 


Circum. 

Area. 


Diam¬ 




eter. 

) 






332853 

701 

2202-26 

385945 

333876 

702 

2205-40 

387047 

334901 

703 

2208-54 

388151 

335927 

704 

2211-68 

389256 

336955 

705 

2214-82 

390363 

337985 

706 

2217-96 

391471 

339016 

707 

2221-11 

392580 

340049 

708 

2224-25 

393692 

341083 

709 

2227-39 

394805 

342119 

710 

2230-53 

395919 

343157 

711 

2233-67 

397035 

344196 

712 

2236-81 

398153 

345237 

713 

2239-96 

399272 

346279 

714 

2243-10 

400393 

347323 

715 

2246-24 

401515 

348368 

716 

2249-38 

402639 

349415 

717 

2252-52 

403765 

350464 

718 

2255-66 

404892 

351514 

719 

2258-81 

406020 

352565 

720 

2261-95 

407150 

353618 

721 

2265-09 

408282 

354673 

722 

2268-23 

409416 

355730 

723 

2271-37 

410550 

356788 

724 

2274-51 

411687 

357847 

725 

2277-65 

412825 

358908 

726 

2280-80 

413965 

359971 

727 

2283-94 

415106 

361035 

728 

2287-08 

416248 

362101 

729 

2290-22 

417393 

363168 

730 

2293-36 

418539 

364237 

731 

2296-50 

4196S6 

365308 

732 

2299-65 

420835 

366380 

733 

2302-79 

421986 

367453 

734 

2305-93 

423139 

368528 

735 

2309-07 

424292 

369605 

736 

2312-21 

425 447 

370684 

737 

2315-35 

426604 

371764 

738 

2318-50 

427762 

372845 

739 

2321-64 

428922 

373928 

740 

2324-78 

430084 

375013 

741 

2327-92 

431247 

376099 

742 

2331-06 

432412 

377187 

743 

2334-30 

433578 

378276 

744 

2337-34 

434746 

379367 

745 

2340-49 

435916 

380459 

746 

2343-63 

437087 

381554 

747 

2346-77 

438259 

382649 

748 

2349-91 

439433 

383746 

749 

2353-05 

440609 

384845 

750 

2356-19 

441786 




















Circumference and Area of Circles. 


69 



Circum. 

Area. 


Circurrt. 

Area. 


Circum. 

Arpa. 

Diam¬ 

eter. 

o 

til 

Diam¬ 

eter. 

o 


Diam¬ 

eter. 

o 

in 

751 

2359-34 

442965 

801 

2516-42 

503912 

851 

2673-50 

568786 

752 

2362-48 

444146 

802 

2519-56 

505171 

852 

2676-64 

570124 

753 

2365-62 

445328 

803 

2522-70 

506432 

853 

2679-78 

571463 

754 

2368-76 

446511 

804 

2525-84 

507694 

854 

2682-92 

572803 

755 

2371-90 

447697 

805 

2528-98 

508958 

855 

2686-06 

574146 

756 

2375-04 

44S883 

806 

2532 12 

510223 

856 

2689-20 

575490 

757 

2378-19 

450072 

807 

2535-27 

511490 

857 

2692-34 

576835 

758 

2381-33 

451262 

808 

2538-41 

512758 

858 

2695-49 

578182 

759 

2384-47 

452453 

809 

2541-55 

514028 

859 

2698-63 

579530 

760 

2387-61 

453646 

810 

2544-69 

515300 

860 

2701-77 

580880 

761 

2390-75 

454841 

811 

2547-83 

516573 

861 

2704-91 

582232 

762 

2393-89 

456037 

812 

2550-97 

517848 

862 

2708-05 

583585 

763 

2397-04 

457234 

813 

2554-11 

519124 

863 

2711-19 

584940 

764 

2400-18 

458434 

814 

2557-26 

520402 

864 

2714-34 

586297 

765 

2403-32 

459635 

815 

2560-40 

521681 

865 

2717-48 

587655 

766 

2406-46 

460837 

816 

2563-54 

522962 

866 

2720-62 

589014 

767 

2409-60 

462041 

817 

2566-68 

524245 

867 

2723-76 

590375 

768 

2412.74 

463247 

818 

2569-82 

525529 

868 

2726-90 

591738 

769 

2415-88 

464454 

819 

2572-96 

526814 

869 

2730-04 

593I02 

770 

2419-03 

465663 

820 

2576-11 

528102 

870 

2733-19 

594468 

771 

2422-17 

466873 

821 

2579-25 

529391 

871 

2736-33 

595835 

772 

2425-31 

468085 

822 

2582-39 

530681 

872 

2739-47 

597204 

773 

2428-45 

469298 

823 

2585-53 

531973 

873 

2742-61 

598575 

774 

2431-59 

470513 

824 

2588-67 

533267 

874 

2745-75 

599947 

775 

2434-73 

471730 

825 

2591-81 

534562 

875 

2748-89 

601320 

776 

2437-88 

472948 

826 

2594-96 

535858 

876 

2752-04 

602696 

777 

2441-02 

474168 

827 

2598-10 

537157 

877 

2755-18 

604073 

778 

2444-16 

475389 

828 

2601-24 

538456 

878 

2758-32 

605451 

779 

2447-30 

476612 

829 

2604-38 

539758 

879 

2761-46 

606831 

780 

2450-44 

477836 

830 

2607-52 

541061 

880 

2764-60 

608212 

781 

2453-58 

479062 

831 

2610-66 

542365 

881 

2767-74 

609595 

782 

2456-73 

480290 

832 

2613-81 

543671 

882 

2770-88 

610980 

783 

2459-87 

481519 

833 

2616-95 

544979 

883 

2774-03 

612366 

784 

2463-01 

482750 

834 

2620-09 

5462S8 

884 

2777-17 

613754 

785 

2466-15 

483982 

835 

2623-23 

547599 

885 

2780-31 

615143 

786 

2469-29 

485216 

836 

2626-37 

548912 

886 

2783-45 

.616534 

787 

2472-43 

486451 

837 

2629-51 

550226 

887 

2786-59 

617927 

788 

2475-58 

487688 

S3 8 

2632-65 

551541 

888 

2789-73 

619321 

789 

2478-72 

488927 

839 

2635-80 

552858 

889 

2792-88 

620717 

790 

2481-86 

490167 

840 

2638-94 

554177 

890 

2796-02 

622114 

791 

2485-00 

491409 

841 

2642-08 

555497 

891 

2799-16 

623513 

792 

2488-14 

492652 

842 

2645-22 

556819 

892 

2802-30 

624913 

793 

2491-28 

493S97 

843 

2648-36 

558142 

893 

2805-44 

626315 

794 

2494-42 

495143 

844 

2651-50 

559467 

894 

2808-58 

627718 

795 

2497-57 

496391 

845 

2654-65 

560794 

895 

2811-73 

629124 

796 

2500-71 

497641 

846 

2657-79 

562122 

896 

2814-87 

630530 

797 

2503-85 

498892 

847 

2660-93 

563452 

897 

2818-01 

631938 

798 

2506-99 

500145 

848 

2664-07 

564783 

898 

2821-15 

633348 

799 

2510-13 

501399 

849 

2667-21 

566116 

899 

2824-29 

634760 

800 

2513-27 

502655 

850 

2670-35 

567450 

900 

2827-43 

636173 


















70 


Circumference and Area of Circles. 



Circum. 

Area. 


Circuin. 

Area. 


Circum. 

Diam¬ 



Diam¬ 



Diam¬ 

o 

eter. 

w 

ISP 

eter. 



eter. 

w 

901 

2830-58 

637587 

934 

2934-25 

685147 

987 

3037-92 

902 

2833*72 

639003 

935 

2937-39 

686615 

968 

3041-06 

903 

2836-86 

640421 

936 

2940-53 

688084 

969 

3044-20 

904 

2840-00 

641840 

937 

2943-67 

689555 

970 

3047-34 

905 

2843-14 

643261 

938 

2946-81 

691028 

971 

3050-49 

906 

2846-28 

644683 

939 

2949-96 

692502 

972 

3053-63 

907 

2849-42 

646107 

940 

2953-10 

693978 

973 

3056-77 

908 

2852-57 

647533 

941 

2956-24 

695455 

974 

3059-91 

909 

2855-71 

648960 

942 

2959-38 

696934 

975 

3063-05 

910 

2858-85 

650388 

943 

2962-52 

698415 

976 

3066-19 

911 

2861-99 

651818 

944 

2965-66 

699897 

977 

3069-34 

912 

2865-13 

653250 

945 

2968-81 

701380 

978 

3072-48 

913 

2868-27 

654684 

946 

2971-95 

702865 

979 

3075-62 

914 

2871-42 

656118 

947 

2975-09 

704352 

9S0 

3078-76 

915 

2874-56 

657555 

948 

2978-23 

705840 

981 

3081-90 

916 

2877-70 

658993 

949 

2981-37 

707330 

982 

3085-04 

917 

2880-84 

660433 

950 

2984-51 

708822 

983 

3088-19 

918 

2883-98 

661874 

951 

2987-65 

710315 

984 

3091-33 

919 

2887.-12 

663317 

952 

2990-80 

711809 

985 

3094-47 

920 

2890-27 

664761 

953 

2993-94 

713307 

986 

3097-61 

921 

2893-41 

666207 

954 

2997-08 

714803 

987 

•3100-75 

922 

2896-55 

667654 

955 

3000-22 

716303 

988 

3103-89 

923 

2899-69 

669103 

956 

3003-36 

717804 

989 

3107-04 

924 

2902-83 

670554 

957 

3006-50 

719306 

990 

3110-18 

925 

2905-97 

672006 

958 

3009-65 

720810 

991 

3113-32 

926 

2909-11 

673460 

959 

3012-79 

722316 

992 

3116-46 

927 

2912-26 

674915 

960 

3015-93 

723823 

993 

3119-60 

928 

2915-40 

676372 

961 

3019-07 

725332 

994 

3122-74 

929 

2918-54 

677831 

962 

3022-21 

726842 

995 

3125-88 

930 

2921-68 

679291 

963 

3025-35 

728354 

996 

3129-03 

931 

2924-82 

680752 

964 

3028-50 

729867 

997 

3132-17 

932 

2927-96 

682216 

965 

3031-64 

731382 

998 

3135-31 

933 

2931-11 

1 683680 

966 

3034-78 

732899 

999 

3138-45 



734417 

735937 

737458 

738981 

740506 

742032 

743559 

745088 

746619 

748151 

749685 

751221 

752758 

754296 

755837 

757378 

758922 

760466 

762013 

763561 

765111 

766662 

768215 

769769 

771325 

772882 

774441 

776002 

777564 

779128 

780693 

782260 

783828 


Explanation of tlie Preceding Table. 

When the diameter is expressed 

in more or less units than in the table, add or 

subtract so many figures more in 

the circumference; 

add or subtract twice as 

many in the area. 

Examples. 


Diameter. 

Circumference. 

Area. 

9370 

29436.7 

68955500 

93.7 

294.367 

6895.55 

9.37 

29.4367 

68.9555 

0.937 

2.94367 

0.689555 
















N YSTROM’S CALCULATOR. 


7 L 


N YSTROM’S CALCULATOR. 



Ail calculations in this Pocket Book have been computed by this Instrument 
It consists of a silvered brass plate on which are fixed two moveable arms, ex¬ 
tending from the centre to the periphery. On the plate are engraved a number 
of curved lines iu such form and divisions, that by their intersection with the 
arms, numbers are read and problems solved. 

The arrangement for trigonometrical calculations is such that it is not neces¬ 
sary to notice sine, cosine, tan, &c., Ac., operating only by the angles them¬ 
selves expressed in degrees and minutes. This makes trigonometrical solutions 
so easy, that any one who understands Simple Arithmetic, will be able to solve 
trigonometrical questions. 

C dculations are performed by it almost instantly, no matter how complicated 
they may be, while there is nothing intricate or difficult in its use. The author 
of this book, who is the inventor, has thoroughly tested its practical utility. 
Without this instrument not one-tenth of the calculations and tables which 
he is continually bringing out, could be produced. 

ADVERTISEMENT. 

The attention of Engineers, Ship builders, and all whose business requires 
frequent and extensive calculations, is called to Nystrom’s Calculator. Brice 
$30. To be obtained with description by applying to John W. Nystrom, Phila¬ 
delphia. 

Communications will be promptly attended to. 

This Calculator received the First Premium at the Franklin Ins>tituto 
Exhibition. 

Wm. J. YOUNG, 

Mathematical, Optical, and Calculating Machine Manufacturer, 

43 N. 7th St., Philadelphia. 





































n 


Circumferences and Areas of Circles. 


Itiame- 

Circ. 

/—x 

Area. 

I 

Circ. 

/■ —\ 

Area 

Diame- 


Circ. 

/ -V 

Area* 

f \ 


Diame- 

( \ 


ter. 


( , ^ 


ter. 

O 

iSp 

ter. 


f§ljf 




WSl' 

3? r 

*0981 

•00076 

5— 

15-70 

19-635 

.11- 


34.55 

95-033 

Vs 

•1963 

.00306 


16-10 

20-629 



34-95 

97-205 

A -- 

•3926 

•01227 

i - 

16-49 

21-647 

i - 


35-34 

99-402 

is 

•5890 

•02761 


16-88 

22-690 



35-73 

101-62 

X 

•7854 

•04908 

A- 

17-27 

23-758 

i— 


36-12 

103-86 

A 

•9817 

•07669 


17-67 

24-850 



36-52 

106-13 

t ~ 

1-178 

•1104 

i - 

18-06 

25-967 

1 n 


36-91 

108-43 

7 

1,5 

1-374 

•1503 


18-45 

27-108 



37-30 

110-75 

A — 

1-570 

•1963 

6 — 

18-84 

28-274 

12— 


37-69 

113-09 

Vs 

1-767 

•2485 


19-24 

29-464 



38-09 

115-46 

t - 

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116- 2 
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120 ' 
120 ' 
121 ' 
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123- 3 

123- 7 

124- 
124-4 

124- 8 

125- 2 

125- 6 

126- 

126- 4 
126-8 

127- 2 

127- 6 

128- 
128-4 


Area. 



962-11 

968-99 

975-90 

982-84 

9S9-80 

996-78 

1003-7 

1010-8 

1017-8 

1024-9 

1032-0 

1039-1 

1046-3 

1053-5 

1060-7 

1067-9 

1075-2 

1082-4 

1089-7 

1097-1 

1104-4 

1111-8 

1119-2 

1126-6 

1134-1 

1141-5 

1149-0 

1156-6 

1164-1 

1171-7 

1179-3 

1186-9 

1194-5 

1202-2 

1209-9 

1217-6 

1225-4 

1233-1 

1240-9 

1248-7 

1256-6 

1264-5 

1272-3 

1280-3 

1288-2 

1296-2 

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131- 9 

132- 3 

132- 7 

133- 1 
133-5 

133- 9 

134- 3 

134- 6 
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135- 4 

135- 8 

136- 2 

136- 6 
137* 

137- 4 

137- 8 

138- 2 

138- 6 

139- 
139-4 

139- 8 

140- 1 
140-5 

140- 9 

141- 3 

141 

142 
142 

142 

143 
143 
144-1 
144-5 

144- 9 

145- 2 

145- 6 

146- 
146-4 

146- 8 

147- 2 


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1328-3 
1336-4 
1344-5 
1352-6 
1360-8 
1369-0 
1377-2 
1385-4 
1393-7 
1401-9 
1410-2 
1418-6 
1426-9 
1435-3 
1443-7 
1452-2 
1460-6 
1469-1 
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1486-1 
1494-7 
1503-3 
1511-9 
1520-5 
1529-1 
1537-8 
1546-5 
1555-2 
1564-0 
15-72-8 
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11599*2 
1608-1 
1617-0 
1625-9 
1634-9 
1643-8 
1652-8 
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148 - 
148-4 

148 - 8 

149 - 2 

149 - 6 

150 - 
150 - 
150 - 

150 - 

151 - 

151- 

152 - 

152- 

153 - 1 
153-5 

153 - 9 

154 - 3 

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155 - 1 
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156 - 6 
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157 - 4 

157 - 8 

158 - 2 

158 - 6 

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159 - 8 

160 - 2 
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161 - 7 

162 - 1 
162-5 

162 - 9 

163 - 3 

163 - 7 

164 - 1 
164-5 

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165 - 3 

165 - 7 

166 - 1 


! Area. 



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1744-1 
1753-4 
1762-7 
1772-0 
1781-3 
1790-7 
1800-1 
1809-5 
1818-9 
1828-4 
1837-9 
1847-4 
1856-9 
1866-5 
1876-1 
1885-7 
1895-3 
1905-0 
1914-7 
1924-4 
1934-1 
1943-9 
1953-6 
1963-5 
1973-3 
1983-1 
1993-0 
2002-9 
2012-8 
2022-8 
2032-8 
2042-8 
2052-8 
2062-9 
2072-9 
2083-0 
2093-2 
2103-3 
2113-5 
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2133-9 
2144-1 
2154-4 
2164-7 
2175-0 
2185-4 
2195-7 

























Circumferences and Areas of Circles. 73 


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ter. 

Circ. 

O 

Area. 

Diame¬ 

ter. 

Circ. 

O 

Area. 

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Circ. 

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185-7 

2745-5 


204-5 

3331-0 

£ 

167-2 

167-6 

2227-0 

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186-1 

186-5 

2757-1 

2768-8 

£ - 

204- 9 

205- 3 

3343-8 

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168- 

168-4 

2248-0 

2258-5 

i— 

186- 9 

187- 3 

2780-5 

2792-2 

i— 

205- 7 

206- 1 

3369-5 

3382-4 

1 - 

168-8 

169-2 

2269-0 

2279-6 

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187- 7 

188- 1 

2803-9 

2815-6 

2 - 

206-5 

206-9 

3395-3 

3408-2 

54— 

169-6 

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2290-2 

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207-3 

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171-2 

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208-9 

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171-6 

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190-4 

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209-3 

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2 - 

172- 

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| - 

190-8 

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209-7 

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172-3 

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191-2 

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210- 

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193-6 

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212-4 

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175-1 

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£ - 

193-9 

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2 - 

212-8 

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194-3 

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213-2 

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1 175*9 

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195-1 

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214- 

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£ - 

176-7 

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2 - 

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197-5 

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216-3 

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179- 

2551-7 

63-4 

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69- 

216-7 

3739-2 


179-4 

2562-9 


198-3 

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217-1 

3752-8 

£ - 

179-8 

2574-1 

£ - 

198-7 

3142-0 

£ - 

217-5 

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218-3 

218-7 

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£ - 

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2 - 

219-1 

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£ - 

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1 






























Circumferences and Areas of Circles. 


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Circ. 

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242-2 

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261-1 

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£ - 

242-6 

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£ - 

261-5 


224-2 

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£ - 

263-1 


225-8 

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244-6 

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263-5 

72-1 

226-1 

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78— 

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263-8 


226-5 

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245-4 

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264-2 

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£ - 

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227-3 

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246-2 

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265- 


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265-4 


228-1 

4142-5 


247- 

4855-2 


265-8 

£ - 

228-5 

4156-7 

£ - 

247-4 

4870-7 

£ 

266-2 


228-9 

4171-0 


247-7 

4886-1 


266-6 

73- 

229-3 

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248-1 

4901-6 

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229-7 

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248-5 

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£ - 

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248-9 

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249-3 

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£ - 

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£ - 

269-3 


-232- 

4286-3 


250-9 

5010-8 


269-7 

74- 

-j 232-4 

4300-8 

80—- 

251-3 

5026-5 

86-- 

-270*1 


4232-8 

4315-3 


251-7 

5042-2 


270-5 

£ - 

- 233-2 

4329-9 

£ - 

-252-1 

5058-0 

£ - 

270-9 


-233-6 

4344-5 


-252-5 

5073-7 


271-3 


4234- 

4359-1 

i— 

-252-8 

5089-5 


271-7 


- 234-4 

4373-8 


-253-2 

5105-4 


272-1 

£ - 

-234-8 

4388-4 

£ - 

-253-6 

5121-2 

£ - 

272-5 


235*2 

4403-1 


-254- 

5137-1 


-272-9 

75- 

-235-6 

4417-8 

81-4 

•254-4 

5153-0 

87— 

273-3 


- 236- 

4432-6 


-254-8 

5168-9 


273-7 

£ - 

- 236-4 

4447-3 

£ - 

-255-2 

5184-8 

£ - 

274-1 


n 236-7 

4462-1 


-255-6 

5200-8 

- 274-4 

£-1 

n 237-l 

4476-9 

i- 

-256- 

5216-8 


- 274-8 


237-5 

4491-8 


256-4 

5232-8 

- 275-2 

£ - 

- 237-9 

4506-6 

£ - 

-256-8 

5248-8 

£ “ 

- 275-6 


- 238-3 

4521-5 


-257-2 

5264-9 


- 276- 

76- 

-238-7 

4536-4 

82- 

-257-6 

5281-0 

88— 

- 276-4 


-239-1 

4551-4 


-258- 

5297-1 


- 276-8 

1 - 

-239-5 

4566-3 

£ - 

-258-3 

5313-2 

£ - 

- 277-2 


- 239-9 

4581-3 


-258-7 

5329-4 


- 277-6 


- 240-3 

4596-3 

i- 

-259-1 

5345-6 

i- 

- 278- 


- 240-7 

4611-3 


-259-5 

15361-8 


r 278-4 

1 

- 241-1 

4626-4 

£ - 

- 259-9 

i 5378-0 

£ * 

- 278-8 

l 241-5 

4641-5 

_ 

r 260-3 

| 5394-3 


- 279-2 


A vaa 



5410-6 
5426-9 
5443-2 
5459-6 
5476-0 
5492-4 
5508-8 
5525-3 
5541-7 
5558-2 
5574-8 
5591-3 
5607-9 
5624-5 
5641-1 
5657-8 
5674-5 
5691 2 
5707-9 
5724-6 
5741-4 
5758-2 
5775-0 
5791-9 
5808-8 
5825-7 
5842-6 
5859-5 
5876-5 
5893-5 
5910-5 
5927-6 
5944-6 
5961-7 
! 597S-9 
1 5996-0 
6013-2 
6030-4 
6047-6 
i 6064-8 
! 6082-1 
: 6099-4 
6116-7 
6134-0 
6151-4 
6168-8 
6186-2 
i6203-6 













































Circumferences and Areas of Circles. 


Circ. 


Diame¬ 

ter. 



89 

a 


90- 

i - 
i- 
S - 

91 - 
i - 
i- 
i - 

82- 
J - 
l- 
* - 


279-6 

279- 9 

280- 3 

280- 7 

281- 1 
281*5 

281- 9 

282- 3 

282- 7 

283- 1 

283-5 

283- 9 
284*3 

284- 7 

285- 1 
285-4 

285- 8 

286- 2 
(-*286-6 
J 287* 

i 287*4 
2S7-8 

288-2 

288-6 

289- 

289-4 

289- 8 
290 2 

290- 5 

290- 9 

291- 3 
U 291-7 


77 


j Area. 

Diame¬ 

ter. 

Circ. 

O 

Area. 

| Diame¬ 
ter. 

Circ. 

o 

! Area. 

6221-1 

6238-6 

93 - 

,292*1 
- 292-5 

6792-9 

6811*1 

97- 

r 304-7 
-305-1 

7389-8 
: 7408-8 

1 6256-1 
! 6273-6 

i ' 

- 292-9 

- 293*3 

6829-4 

6847-8 

i - 

-305*5 

-305-9 

7427-9 

7447-0 

6291-2 


- 293-7 

6866-1 


- 306-3 

7466-2 

1 6308-8 

-* 294-1 

1 6884-5 


- 306-6 

7485-3 

6326-4 

£ _ 
4 

294-5 

1 6902-9 

1 - 

307- 

7504*5 

6344-0 


- 294-9 

6921-3 

307-4 

7523-7 

6361-7 

94— 

- 295-3 

6939-7 

98- 

- 307-8 

7542-9 

6379*4 


- 295-7 

6958-2 


- 308-2 

7562-2 

6397-1 

1 - 
4 

- 296- 

6976-7 

i - 

308-6 

7581-5 

6414*8 


- 296-4 

6995-2 


309-0 

7600-8 

6432-6 

4- 

- 296-8 

7013-8 

i- 

309-4 

7620-1 

6450-4 


- 297-2 

7032-3 


309-8 

7639-4 

6468-2 

s - 

297-6 

7050-9 

s 

310*2 

7658-8 

6486 0 


- 298- 

7069-5 


310-6 

7678*2 

! 6503-8 

95 — 

298-4 

7088-2 

99- 

311*0 

7697-7 

i 6521-7 


- 298-8 

7106-9 


311-4 

7717-1 

6539-6 

1 - 
4 

-299-2 

7125-5 

i - 

311-8 

7736-6 

6557-6 


^299-6 

7144-3 

312-1 

7756-1 

6575-5 

i- 

1300- 

7163-0 

i-- 

312-5 

7775-6 

6593-5 


-300-4 

7181-8 


312-9 

7795-2 

6611-5 

s - 

-300-8 

7200-5 

1 - 

313-3 

7814-7 

6629-5 


-301*2 

7219-4 


313 7 

7834-3 

6647-6 

96— 

-301*5 

7238-2 

100-- 

314-1 

7853-9 

6665-7 


-301-9 

7257-1 


314-5 

7853-6 

6683-8 

i - 

302-3 1 

7275-9 

i -- 

314-9 

7893-3 

6701-9 


-302-7 

7294-9 


315-3 

7913-1 

6720-0 

i— 

303-1 

7313-8 

i- 

315-7 

7932-7 

6738-2 


303-5 

7332-8 


316-0 

7942-4 

6756-4 

| -- 

303-9 ! 

7351-7 

2 - 

316-4 

7972-2 

6776-4 


304-3 j 

7370-7 


316-8 

7991-9 


i 

i 


i 


EXPLANATION OP THE TABLE FOR SEGMENTS, &c. 

The chord divided by the height is the gauge in the Table, the quotient in the 
first column. 

k = tabular coefficient, always to be multiplied by the chord. 

To find the angle of an arc of a circle* 

RULE. Divide the base (chord) of the arc by its height, (sine verse ) and find I 
the quotient in the first column. The corresponding number in the second 
column is the angle of the arc in degrees of the circle. 

To find the radius of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in j 
! the first column. The corresponding number in the third oolumn, multiplied 
I by the chord, is the radius of the arc. 










































78 lABLE FOR SEG3IENT3 &C., OF A CIRCLE. 


Chord div. 

Centre 

Radius 

Cir. Arc. 

Area Seg. 

Surface 

Solidity 

- - - ' " - 

Chord 

ly beigbt. 

Angle v. 

r = 

h c. 

1 k c- 

( 7 ; =x h C2. 

a = k c a 

. C= A e® 

. c = hr. 



'"A 

< 

r —^ 





1 '' "7s 



N 

r» 




Y 

7 H / 

X'' 

458-08 

1 

57-296 

1-0000 

•01091 

•78539 

•000S5 

•01744 

229-18 

2 

28-649 

1-0000 

•00218 

•78549 

•00172 

•03490 

152-77 

3 

19-101 

1-0000 

•00327 

•78462 

•00255 

•05234 

114-57 

4 

14-327 

1-0000 

•00436 

•78574 

•00310 

•06978 

84-747 

5 

11-462 

1-0001 

.00647 

•78586 

•00401 

•08722 

76-375 

6 

9-5530 

1-0003 

•00741 

•78599 

•00514 

•10466 

65-943 

7 

8-1902 

1-0004 

•00910 

‘•78621 

•00592 

•12208 

57-273 

8 

7-1678 

1-0006 

•010S9 

•78630 

•00686 

•13950 

50-902 

9 

6-3728 

1-0008 

•01254 

•78665 

•00772 

•15690 

45-807 

10 

5-7368 

1-0011 

•01407 

•78695 

•00857 

•17430 

41-203 

11 

5-2167 

1-0013 

•01552 

•78730 

•00964 

•19168 

38-133 

12 

4-7834 

1-0016 

•01695 

•78725 

•01031 

1 -20904 

35-221 

13 

4-4168 

1-0019 

•01841 

•78794 

•01114 

1 -22640 

32-742 

14 

4-1027 

1-0023 

•02000 

•78832 

•01199 

•24372 

30-514 

15 

3-8307 

1-0027 

•02157 

•78889 

•01288 

•26! 04 

28-601 

16 

3-5927 

1-0029 

•02269 

•78909 

•01375 

•27834 

26-915 

17 

3-3827 

1-0034 

•02434 

•78969 

•01462 

•29560 

25-412 

18 

3-1962 

1-0039 

•02592 

•79028 

•01542 

•31286 

24-068 

19 

3-0293 

1-0044 

•02744 

•79084 

•01635 

•33008 

22-860 

20 

2-8793 

1-0048 

•02878 

•79140 

•01722 

•34728 

21-760 

21 

2-7440 

1-0054 

•03040 

•79234 

•01S02 

•36446 

20-777 

22 

2-6222 

1-0059 

•03178 

•79300 

•01897 

•38160 

19-S62 

23 

2-50S0 

1-0066 

*03343 

•79340 

•01984 

•39S72 

19-028 

24 

2-4050 

1-0072 

•03493 

•79416 

•02072 

•41582 

18-261 

25 

2-3101 

1-0078 

•03639 

•79486 

•02159 

•43286 

. 17-553 

26 

2-2233 

1-0084 

•03784 

•79530 

•02248 

•44990 

16-970 

27 

2-1418 

1-0091 

•03970 

•79639 

•02315 

•46688 

16-288 

28 

2-0673 

1-0101 

•04115 

•79748 

•02424 

•48384 

15-721 

29 

1-9969 

1-0105 

•04230 

•79811 

•02511 

•50076 

15-191 

30 

1-9319 

1-0113 

•04385 

•79907 

•02600 

•51762 

14-970 

31 

1-8710 

1-0121 

•04476 

•78530 

•02692 

•53446 

14-230 

32 

1-8140 

1-0129 

•01710 

•80098 

•02778 

•55126 

13-796 

33 

1-7605 

1-0138 

•04842 

•80181 

•02866 

•56802 

13-382 

34 

1-7102 

1-0146 

•04989 

•80300 

•02956 

.58479 

12-994 

35 

1-6628 

1-0155 

•05137 

•80405 

•03046 

•60140 

12-733 

36 

1-6184 

1-0167 

•05311 

•80531 

•03137 

•61S02 

12-473 

37 

1-5758 

1-0174 

•05401 

•80622 

•03226 

•63460 

11-931 

38 

1-5358 

1-0184 

•05628 

•80713 

•03328 

•65112 

11-621 

39 . 

1-4979 

1-0194 

•05755 

•80850 

•03418 

•66760 

11-342 

40 

1-4619 

1-0204 

•05899 

•809S7 

•03506 

•68404 

11-060 

41 

1-4266 

1-0207 

•06001 

•81046 

•03589 

•70040 

10-791 

42 

1-3952 

1-0226 

•06196 

•81240 j 

•03680 

•71672 

10-534 

43 

1-3643 

1-0237 

•06359 

•81377 

•03773 

•73300 

10-289 

44 

1-3347 

1-0248 

•06574 

•81505 ; 

•03864 

•74920 

10-043 

45 

1-3 066 

1-0260 

•06628 

•81756 | 

•03890 

•76536 

9-8303 

46 

1-2797 

1-0272 

•06826 

.81795 

•01050 

•78146 

9-6153 

47 

1-2539 1 

1-0290 

•06998 

•81939 

•04143 

•79748 | 

9-4092 

48 

1-2289 | 

1-0297 

•09138 ! 

1 

i 

•82064 

•04247 j 

•81346 ! 

| ; 



















































Table for Segments &c., of a Circle. 


79 


Chord dir 

Centre 

Rad ius 

Cir. Arc. 

Area Srg. 

Surface 

Solidity 

Chord 

by height. 

Angle v. 

r = ft c. 

b—kc. 

a — A e®. 

a = h c 2 - 

C — k c 3 . 

c = A r. 





A 

rffmnjlKaHwv 



V'' 





N ^ 

'v' 

9-2113 

49 

1-2057 

1-0309 

•07290 

•82244 

•04330 

•82938 

9-0214 

50 

1-1831 

1-0323 

•07453 

•82384 

•04424 

•84522 

8-8387 

51 

1-1614 

1-0336 

•07611 

•82562 

•04519 

•86102 

8-6629 

52 

1*1406 

1-0349 

•07758 

•82729 

•04614 

•87674 

8-4462 

53 

1-1206 

1-0364 

•07959 

•83363 

•04685 

•89238 

8-3306 

54 

1-1014 

1-0378 

•08083 

•83072 

•04805 

•90798 

8 1733 

55 

1-0828 

1-0393 

•08246 

•83249 

•04901 

•92348 

8 0215 

56 

1-0650 

1-0407 

•08400 

•83422 

•05002 

•93894 

7-8750 

57 

1-0478 

1-0422 

•08579 

•83602 

•05098 

•95430 

7-7334 

58 

1-0313 

1-0431 

•08680 

•83796 

•05191 

•96960 

7-5895 

59 

1-0154 

1-0454 

•08S91 

•84064 

•05299 

•98484 

7-4565 

60 

1-0000 

1-0470 

•09106 

•84266 

•05400 

1-0000 

7'3358 

61 

•98515 

1-0486 

•09209 

•84380 

•054P6 

1-0150 

7-2118 

62 

•97080 

1-0503 

•09375 

•84581 

•05583 

1-0300 

7-0914 

63 

•95694 

1-0520 

•09540 

•84791 

•05684 

1-0450 

6-9748 

64 

•94352 

1-0537 

•09697 

•84996 

•05784 

1-0598 

6-S616 

65 

•93058 

1-0555 

•09865 

•85215 

•05885 

1-0746 

6-7512 

66 

•91804 

1*0573 

•10036 

•85441 

•05987 

1-0S92 

6-6453 

67 

•90590 

1-0591 

•10201 

•S5640 

•06088 

1-1038 

6-5469 

68 

•89415 

1-0610 

•10367 

•85815 

•06181 

1-1184 

6*4902 

69 

•88276 

1-0629 

•10520 

•85464 

•06201 

1-1328 

6-3431 

70 

•87172 

1-0648 

•10710 

•86350 

•06396 

1-1471 

6-2400 

71 

•86102 

1-0668 

•10887 

•86699 

•06515 

1-1614 

6-1553 

72 

•85065 

1-0687 

•11046 

•86834 

•06604 

1-1755 

6-0652 

73 

•84058 

1-0708 

•11225 

•87081 

•06709 

1*1896 

5-9773 

74 

•83082 

1-0728 

•11385 

•87935 

•06815 

1-2036 

5-8918 

75 

•82134 

1-0749 

•11563 

•87590 

•06921 

1 *2175 

5-8084 

76 

•81213 

1-0770 

•11736. 

•87853 

•07037 

1-2313 

5-7271 

77 

•80319 

1-0792 

•11910 

•88120 

•07136 

1-2450 

5-6478 

78 

•79449 

1-0814 

•12072 

•88389 

•07244 

1-2586 

5-5704 

79 

•78606 

1-0836 

•12281 

•88677 

•07352 

1-2721 

5*4949 

80 

•77786 

1-0859 

•12441 

•88949 

•07462 

l*285f 

5-4254 

81 

•76988 

1-0882 

•12660 

•S9161 

•07512 

1-2989 

5-3492 

82 

•76212 

1-0905 

•12793 

•S9520 

•07683 

1-3121 

5-2705 

83 

•75458 

1-0920 

•12958 

•89958 

•07819 

1-3252 

5-2101 

84 

•74724 

1-0953 

•13157 

•90095 

•07907 

1-3383 

5-1429 

85 

•74009 

1-0977 

•13330 

•90420 

•07960 

1-3512 

5-0772 

86 

•73314 

1-1012 

•13546 

•90734 

•08102 

1-3639 

5-0134 

87 

•72637 

1-1027 

•13704 

•91036 

•0S340 

1-3767 

4-9501 

88 

•71978 

1-1054 

•13893 

•91363 

•08436 

1-3893 

4-8886 

S9 

•71336 

1-1079 

•14078 

•91696 

•08530 

1-4818 

4-8216 

90 

•70710 

1-1105 

•14279 

•92210 

•0S621 

1-4142 

4-7694 

91 

•70101 

1-1132 

•14449 

•92352 

•08716 

1-4205 

4-7117 

92 

•69508 

1-1159 

•14643 

•92476 

•0S798 | 

1-4387 

4 c b615 

93 

•68930 

1-1186 

•14817 

•92914 

•08932 

1-4507 

4-5999 

94 

•68366 

J-1211 

•15009 

•93385 

•09076 

1-4627 

4-5453 | 

95 

•67817 

1-1242 

•15211 

•93746 

•09197 1 

1-4745 

4-4845 1 

96 

•67282 

1-1271 

•15375 

•94272 

•09348 1 

1-4S63 












































Table for Sfrmextj &c., or a Circle. 


Chord div 

Centre 

Rad ius 

Cir. Arc. 

Area Sep. 

Surface 

Solidity 

Chord 

bv height. 

Angle v. 

r = 

k c. 

4 = lc. 

a = A o a . 

C»* 

II 

e 

© == A c*. 

e = k r. 


/.Vs''. 

V 

/ 

N 

r ? 

s 







\ 

N 

\ / 





4-4398 

97 

•66760 

1-1300 

•15600 

•94470 

•09442 

1-4979 

4-3859 

98 

•66250 

1-1329 

•15801 

•94852 

•09567 

1-5094 

4-3383 

99 

•65754 

1-1359 

•15995 

•95236 

•09693 

1-5208 1 

4-2862 

100 

•65270 

1-1382 

•16180 

•95682 

•09831 

1-5321 

4-2406 

101 

•64798 

1-1420 

.16393 

•96011 

•09856 

1-5432 

4-1930 

102 

•64338 

1-1451 

■16610 

•96412 

•10076 

1-5543 

4-1570 

103 

•63889 

1-1483 

•16925 

•96568 

•10557 

1-5652 

4-1006 

104 

•63450 

1-1515 

•17001 

•97246 

•10273 

1-5760 

4-0555 

105 

•63023 

1-15.47 

•17204 

•97643 

•10471 

1-5867 

4-0113 

106 

•62607 

1-1580 

•17414 

•98067 

•10601 

1-5973 

3-9679 

107 

•62200 

1-1614 

•17619 

•98495 

•10735 

*1-6077 

3-9252 

108 

•61803 

1-1648 

•17832 

•98931 

•10870 

1-6180 

3-8832 

109 

•61416 

1-1682 

•18041 

•99376 

•11007 

1-6282 

3*8419 

110 

•61039 

1-1716 

•18257 

•98827 

•11149 

1-6383 

3-S013 

111 

•60670 

1-1752 

•18472 

1-0028 

•11284 

1-6482 

3-7612 

112 

•60325 

1-1790 

•18696 

1-0077 

•11426 

1-6581 

3-7221 

113 

•59960 

1-1823 

•18900 

1-0122 

•11566 

1-6677 

3-6837 

114 

•59618 

1-1859 

•19117 

1-0169 

•11709 

1-6773 

3-6454 

115 

•59284 

1-1897 

•19339 

1-0218 

•11853 

1-6867 

3-6086 

116 

•58959 

1-1934 

•19559 

1-0266 

•11995 

1-6961 

3-5712 

117 

•58641 

1-1972 

•19787 

1-0317 

•12145 

1-7053 

3-5349 

118 

•58331 

1-2011 

•20009 

1-0368 

•12294 

1-7143 

3-4992 

119 

•58030 

1-2050 

•20227 

1-0417 

•12444 

1-7232 

3-4641 

120 

•57735 

1-2089 

•20453 

1-0472 

•12596 

1-7320 

3-4296 

121 

•57450 

1-2130 

•20678 

1-0525 

•12748 

1-7407 

3-3953 

122 

•57168 

1-2177 

•20945 

1-0578 

•12903 

1-7492 

3-3616 

123 

•56895 

1-2213 

•21175 

1-0634 

•13060 

1-7576 

3-3285 

124 

•56628 

1-2253 

' -21399 

1-0690 

•13218 

1-7659 

3-2940 

125 

•56370 

1-2295 

•21538 

1-0753 

•13391 

1-7740 

3-2637 

126 

•56116 

1-2338 

•21859 

1-0803 

•13558 

1-7820 

3-2319 

127 

•55870 

1-2381 

•22121 

1-0862 

•13701 

1-7898 

3-2006 

128 

•55630 

1-2425 

•22370 

1-0921 

•13866 

1-7976 

3-1716 

129 

•55396 

1-2470 

•22617 

1-0974 

•14028 

1-8051 

3-1393 

130 

•55169 

1-2515 

•22865 

1-1040 

•14202 

1-8126 

3-1093 

131 

•54947 

1-2561 

•23113 

1-1104 

•14371 

1-8199 

3-0805 

132 

•54732 

1-2607 

•23372 

1-1164 

•14537 

1-8271 

3-0555 

133 

•54522 

1-2654 

•23603 

1-1212 

•14676 

1-8341 

3-0216 

134 

•54318 

1-2701 

•23892 

1-1295 

•14894 

1-8410 

2-9777 

135 

•54120 

1-2749 

•24198 

1-1420 

•15209 

1-8477 

2-9651 

136 

•53927 

1-2798 

•24364 

M428 

•15252 

1-S543 

2-9374 

137 

•53740 

1-2847 

•24676 

1-1495 

•15422 

1-8608 

2-9115 

138 

•53557 

1-2897 

•21938 

1-1558 

•15605 

1-8671 

2-8829 

139 

•53380 

1-2948 

•25222 

1-1634 

•15807 

1-8733 

2-8562 

140 

•53209 

1-2999 

•25485 

1-1705 

•15996 

1-8794 

2-8299 

141 

•53042 

1-3051 

•25759 

1-1777 

•16201 

1-8853 

2-8038 

142 

•52881 

1-3065 

•25936 

1-1851 

•16381 

1-8910 

2-7781 

143 

•52724 

1-3157 

•26320 

1-1925 

•16577 

1-8966 

2-7527 

144 

•52573 

1-3211 

•26604 

1-2000 

•16776 

1-9021 














































Table for Segments &o., op a Circle. gi 


Chord dir. 

Centre 

Rad ms 

Cir. Arc. 

Area Seg. 

Surface 

Solidity 

Chord 

by bright. 

Angle v. 

r = 

kc. 

b = kc. 

a = k c a . 

a = k c 3 - 

C — k c*. 

e = k r. 

<f^> 



7 ~»w 

N 

V > 



i(gm> 


J — x 

V 

XT' 

N 

'/ 

'•a 


Nx" 

\ / 


2*7276 

145 

•52426 

1*3265 

•26889 

1-2077 

*16965 

1-9074 

2*7002 

146 

*52284 

1-3320 

•27196 

1-2166 

*17209 

1-9126 

2*6816 

147 

*52147 

1-3377 

•27449 

1-2219 

*17205 

1*9176 

2-6533 

148 

*52015 

1-3433 

•27772 

1*2318 

•17605 

1*9225 

2*6301 

149 

*51887 

1-3491 

•28168 

1*2396 

*17809 

1*9272 

2-6064 

150 

•51764 

1-3549 

•28369 

1*2476 

*18023 

1*9318 

2-5830 

151 

•51645 

1-3608 

•28674 

1*2563 

*18666 

19363 

2*5598 

152 

*51530 

1-3668 

•28983 

1*2648 

*18751 

1*9406 

2*5239 

153 

•51420 

1-3729 

•29397 

1*2801 

*18845 

1*9447 

2-5143 

154 

•51315 

1*3790 

•29607 

1*2824 

*18913 

1*9487 

2*4919 

155 

•51214 

1-3852 

•29928 

1*2914 

*19147 

1*9526 

2*4699 

156 

•51117 

1-3919 

•30259 

1*3004 

*19374 

1*9563 

2*4478 

157 

•51014 

1*3973 

•30560 

1*3094 

*19607 

1*9598 

2*4262 

158 

•50936 

1-4043 

•30905 

1*3191 

*20029 

1*9632 

2*4047 

159 

•50851 

1-4109 

•31239 

1*3287 

*20095 

1*9663 

2*3835 

160 

•50771 

1-4175 

•31575 

1*3368 

•20342 

1*9696 

2*3613 

161 

•50695 

1-4243 

•31931 

1*3490 

•20609 

1*9725 

2*3417 

162 

•50623 

1-4311 

•32263 

1*3583 

•20847 

1*9753 

2*3211 

163 

•50555 

1-4380 

•32618 

1*3682 

•21105 

1*9780 

2*3004 

164 

•50491 

1-4450 

•32969 

1*3791 

•21371 

1*9805 

2*2805 

165 

•50431 

1-4520 

•33327 

1*3895 

•21634 

1*9829 

2*2605 

166 

•50374 

1-4592 

•33684 

1*4021 

•21904 

1*9851 

2*2408 

167 

•50323 

1-4665 

•34048 

1*4111 

•22177 

1*9871 

2*2212 

168 

•50275 

1-4739 

•34422 

1*4222 

•21946 

1*9890 

2*2013 

169 

•50231 

1-4813 

•34802 

1*4344 

•22766 

1*9908 

2*1826 

170 

•50191 

1-4889 

•35230 

1*4476 

•23028 

1*9924 

2*1636 

171 

•50154 

1-4966 

•35563 

1*4565 

•23266 

1*9938 

2*1447 

172 

•50122 

1-5044 

•35953 

1*4684 

•23650 

1*9951 

212x71 

173 

•50093 

1-5123 

•36337 

1*4797 

•23900 

1*9962 

2*1075 

174 

•50068 

1-5202 

•36747 

1*4927 

•24225 

1*9972 

2-0S92 

175 

•50047 

1-5283 

•37152 

1*5052 

•24537 

1*9981 

2*0710 

176 

•50030 

1-5365 

•37562 

1*5179 

•24856 

1*9988 

2*0530 

177 

•50017 

1-5448 

•37974 

1*5308 

•25179 

1*9993 

2*0352 

178 

•50007 

1-5533 

•38401 

1*5439 

•25531 

1*9996 

2*0175 

179 

•50002 

1-5618 

•38828 

1*5573 

•25840 

1*9999 

2*0000 

180 

•50000 

1-5707 

•39269 

1*5708 

•26179 

2*0000 


To find the length of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in 
the first column. The corresponding number in the fourth eoluinn multiplied 
by the chord is the length of the arc. 


To find the area of a segment of a circle, 

! RULE. Divide the chord of the segment by its height, and find the quotient 
\ In the first column. The corresponding number in the fifth column multiplied 
j hy the square of the chord, is the area of the segment. 

i 


6 













































32 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

\f Roots. 

\/ Roots. 

1 

1 

1 

1-0000000 

1-0000000 

2 

4 

8 

1*4142136 

1-2599210 

3 

9 

27 

1-7320508 

1-4422496 

4 

16 

64 

2-0000000 

1-5874011 

5 

25 

125 

2-23606SO 

1-7099759 

6 

36 

216 

2-4494897 

1-8171206 

7 

49 

343 

2-6457513 

1-9129312 

8 

64 

512 

2-8284271 

2-0000000 

9 

81 

729 

3-0000000 

2-0800837 

10 

100 

1000 

3-1622777 

2-1544347 

11 

121 

1331 

3-3166248 

2-2239801 

12 

144 

1728 

3-4641016 

2-2894286 

13 

169 

2197 

3-6055513 

2-3513347 

14 

196 

2744 

3-7416574 

2-4101422 

15 

225 

3375 

3-8729833 

2-4662121 

16 

256 

4096 

4-0000000 

2-5198421 

17 

289 

4913 

4-1231056 

2-5712816 

18 

324 

5832 

4-2426407 

2-6207414 

19 

361 

6859 

4-3588989 

2-6684016 

20 

400 

8000 

4-4721360 

2-7144177 

21 

441 

9261 

4-5825757 

2-7589243 

22 

484 

10648 

4-6904158 

2-8020393 

23 

529 

12167 

4-7958315 

2-8438670 

24 

576 

13824 

4-8989795 

2-8844991 

25 

625 

15625 

5-000000*0 

2-9240177 

26 

676 

17576 

5-0990195 

2-9624960 

27 

729 

19683 

5-1961524 

3-0000000 

28 

784 

21952 

5-2915026 

3-0365889 

29 

841 

24389 

5-3851648 

3-0723168 

30 

900 

27000 

5-4772256 

3-1072325 

31 

961 

29791 

5-5677644 

3-1413806 

32 

1024 

32768 

5-6568542 

3-1748021 

33 

1089 

35937 

5-7445626 

3-2075343 

34 

1156 

39304 

5-8309519 

3-2396118 

35 

1225 

42875 

5-9160798 

3-2710663 

36 

1296 

46656 

6-0000000 

3-3019272 

37 

1369 

50653 

6-0827625 

3-3322218 

38 

1444 

54872 

6-1644140 

3-3619754 

39 

1521 

59319 

- 6-2449980 

3-3912114 

40 

1600 

64000 

6-3245553 

3-4199519 

41 

1681 

68921 

6-4031242 

3-4482172 

42 

1764 

74088 

6-4807407 

3-4760266 

43 

1849 

79507 

6-5574385 

3-5033981 

44 

1936 

85184 

6-6332496 

3-5303483 

45 

2025 

91125 

6-7082039 

3-5568933 

46 

2116 

97336 

6-7823300 

3-5830479 

47 

2209 

103823 

6-8556546 

3-608S261 

48 

2304 

110592 

6-9282032 

3-6342411 

49 

2401 

117649 

7-0000000 

3-6593057 

50 

2500 

125000 

7-0710678 

3-6840314 

51 

2601 

132651 

7-1414284 

3-7084298 

52 

2704 

140608 

7-2111026 

3-7325111 


Reciprocals. 

1.0000000C 

•500000000 

*333333333 

•250000000 

•200000000 

•166666667 

•142857143 

•125000000 

•111111111 

•100000000 

•090909091 

•083333333 

•076923077 

•071428571 

•066666667 

•062500000 

.058823529 

•055555556 

•052631579 

•050000000 

•047619048 

•045454545 

•043478261 

•041666667 

•040000000 

•038461538 

•037037037 

•035714286 

•034482759 

•033333333 

•032258065 

•031250000 

•030303030 

•029411765 

•028571429 

•027777778 

•027027027 

•026315789 

•025641026 

•025000000 

•024390244 

•023809524 

•023255814 

•022727273 

•022222222 

•021739130 

•021276600 

•020833333 

•020408163 

•020000000 

•019607843 

•019230769 


















Table of Squares, Cubes, Square and Cube Boots, 


m 


Number. 





--- 

Squares. 

Cubes. 

V Roots. 

yj Roots. 

Reciprocals. 

53 

2809 

148877 

7-2801099 

3-7562858 

018867925 

54 

2916 

157464 

7-3484692 

3-7797631 

•018518519 

55 

3025 

166375 

7*4161985 

3-8029525 

•018181818 

56 

3136 

175616 

7*4833148 

3-8258624 

*017857143 

57 

3249 

185193 

7-5498344 

3-8485011 

*017543860 

58 

3364 

195112 

7*6157731 

3*8708766 

•017241379 

59 

3481 

205379 

7-6811457 

3*8929965 

*016949153 

60 

3600 

216000 

7*7459667 

3-9148676 

*016666667 

61 

3721 

226981 

7-8102497 

3-9304972 

*016393443 

62 

3844 

238328 

7-8740079 

3*9578915 

•016129032 

63 

3969 

250047 

7-9372539 

3-9790571 

•015873016 

64 

4096 

262144 

8-0000000 

.4-0000000 

•015625000 

65 

4225 

274625 

8-0622577 

4-0207256 

*015384615 

66 

4356 

287496 

8-1240384 

4-0412401 

•015151515 

67 

4489 

300763 

8-1853528 

4-0615480 

•014925373 

6S 

4624 

314432 

8*2462113 

4*0816551 

•014705882 

69 

4761 

328509 

8*3066239 

4*1015661 

•014492754 

70 

4900 

343000 

8-3666003 

4*1212853 

•014285714 

71 

5041 

357911 

8*4261498 

4-1408178 

•014084517 

72 

5184 

373248 

8-4852814 

4-1601676 

•013888889 

73 

5329 

389017 

8-5440037 

4-1793390 

*013698630 

74 

5476 

405224 

8-6023253 

4*1983364 

*013513514 

75 

5625 

421875 

8-6602540 

4-2171633 

*013333333 

76 

5776 

438976 

8-7177979 

4*2358236 

*013157895 

77 

5929 

456533 

8-7749644 

4*2543210 

*012987013 

78 

6084 

474552 

8-8317609 

4-2726586 

•012820513 

79 

6241 

493039 

8-8881944 

4-2908404 

*012658228 

80 

6400 

512000 

8-9442719 

4*3088695 

*012500000 

81 

6561 

531441 

9-0000000 

4-3267487 

•012345679 

82 

6724 

551368 

9-0553851 

4-3444815 

*012195122 

83 

6889 

571787 

9-1104336 

4-3620707 

*012048193 

84 

7056 

592704 

9-1651514 

4-3795191 

•011904762 

85 

7225 

614125 

9-2195445 

4-3968296 

*011764706 

86 

7396 

636056 

9*2736185 

4-4140049 

•011627907 

87 

7569 

658503 

9*3273791 

4-4310476 

•011494253 

88 

7744 

681472 

9-3808315 

4-4479692 

•011363636 

89 

7921 

704969 

9-4339811 

4-4647451 

*011235955 

90 

8100 

729000 

9-4868330 

4-4814047 

*011111111 

91 

8281 

753571 

9-5393920 

4-4979414 

*010989011 

92 

8464 

778688 

9-5916630 

4-5143574 

•010869565 

93 

- 8649 

804357 

9-6436508 

4-5306549 

•010752688 

94 

8836 

830584 

9-6953597 

4-5468359 

•010638298 

95 

9025 

857375 

9-7467943 

4-5629026 

•010526316 

96 

9216 

884736 

9-7979590 

4-5788570 

•010416667 

97 

9409 

912673 

9-8488578 

4-5947009 

•010309278 

98 

9604 

941192 

9-8994949 

4-6104363 

•010204082 

99 

9801 

970299 

9-9498744 

4-6260650 

•010101010 

100 

10000 

1000000 

10-0000000 

4-6415888 

•010000000 

101 

10201 

1030301 

10-0498756 

4-6570095 

•009900990 

102 

10404 

1061208 

10-0995049 

4-6723287 

009803922 

103 

10609 

1092727 

10-1488916 

4*6875482 

•009708738 

104 

10816 

1324864 

10*1980390 

4-7025694 

•009615385 



















Table of Squares, Cubes,* Square and Cube Roots 


Number. 

Squares. 

Cubes. 

Roots. 

;5 - 

V Roots. 

Reciprocals. 1 

105 

11025 

1157625 

10-2469508 

4-7176940 

•0095° o 810 

106 

11236 

1191016 

10-2956301 

4-7326235 

•009433962 

107 

11449 

1225043 

10-3440804 

4-7474594 

•009345794 

108 

11664 

1259712 

10-3923048 

4-7622032 

•009259259 

109 

11881 

1295029 

10-4403065 

4-7768562 

•009174312 

110 

12100 

1331000 

10-4880885 

4-7914199 

•009090909 

111 

12321 

1367631 

10-5356538 

4-805S995 

•009009009 

112 

12544 

1404928 

10-5830052 

4-8202845 

•008928571 

113 

12769 

1442897 

10-6301458 

4-8345881 

•008849558 

114 

12996 

1481544 

10-6770783 

4-8488076 

•008771930 

115 

13225 

1520875 

10-7238053 

4-8629442 

•008695652 

116 

13456 

1560896 

10-7703296 

4-8769990 

•008620690 

117 

13689 

1601613 

10-8166538 

4-8909732 

•008547009 

118 

13924 

1643032 

10-8627805 

4-9048681 

•008474576 

119 

14161 

1685159 

10-9087121 

4-9186847 

•008403361 

120 

14400 

1728000 

10-9544512 

4-9324242 

•008333333 

121 

14641 

1771561 

11-0000000 

4-9460874 

•008264463 

122 

14884 

1815848 

11-0453610 

-4-9596757 

•008196721 

123 

15129 

1860S67 

11-0905365 

4-9731898 

•008130081 

124 

15376 

1906624 

11-1355287 

4-9866310 

•008064516 

l 125 

15625 

1953125 

11-1803399 

5-0000000 

•008000000 

126 

15876 

2000376 

11-2249722 

5-0132979 

•007936508 

127 

16129 

20483S3 

11-2694277 

5-0265257 

•007874016 

128 

[ 129 

16384 

2097152 

11-3137085 

5-0396842 

•007812500 

16641 

2146689 

11-3578167 

5-0527743 

•007751938 

130 

16900 

2197000 

11-4017543 

5-0657970 

•007692308 

131 

17161 

2248091 

11-4455231 

5-0787531 

•007633588 

132 

17424 

2299968 

11-4891253 

5-0916434 

•007575758 

133 

17689 

2352637 

11-5325626 

5-1044687 

•007518797 

134 

17956 

2406104 

11-5758369 

5-1172299 

•007462687 

135 

18225 

2460375 

11-6189500 

5-1299278 

•007407407 

136 

18496 

2515456 

11-8619038 

5-1425632 

•007352941 

137 

18769 

2571353 

11-7046999 

5-1551367 

•007299270 

138 

19044 

262S072 

11-74734 01 

5-1676493 

•007246377 

139 

19321 

2685619 

11-7898261 

5-1801015 

•007194245 

140 

19600 

2744000 

11-8321596 

5-1924941 

•007142857 

141 

19S81 

2803221 

11-S743421 

5-2048279 

•007092199 

142 

20164 

2863288 

11-9163753 

5-2171034 

•007042254 

143 

20449 

2924207 

11-9582607 

5-2293215 

•006993007 

144 

20736 

2985984 

12-0000000 

5-2414828 

•006944444 

145 

21025 

3048625 

12-0415946 

5-2535879 

•006896552 

146 

21316 

3112136 

12-0S30460 

5-2656374 

•006849315 

147 

21609 

3176523 

12-1243557 

5-2776321 

•006802721 

148 

21904 

3241792 

12-1655251 

5-2895725 

•006756757 

149 

22201 

3307949 

12-2065556 

5-3014592 

•006711409 

150 

22500 

3375000 

12-2474487 

5-3132928 

•006666667 

151 

22801 

3442951 

12-2882057 

5-3250740 

•006622517 

152 

23104 

3511008 

12-3288280 

5-3368033 

•006578947 

153 

23409 

3581577 

12-3693169 

5-34S4812 

•006535948 

154 

23716 

3652264 

12-4096736 

5-3601084 

•006493506 

155 

24025 

3723875 

12-4498996 

5-3716854 

•006451613 

156 

24336 

3796416 

12-4899960 

5-3832126 

•006410256 






















Table of Squares, Cubes, Square and Cube Roots. 


85 


Number. 

Squares. 

Cubes. 

\l Roots. 

4/ Roots. 

Reciprocals. 

157 

24649 

3869893 

12-5299641 

5-3946907 

•006369427 

158 

24964 

3944312 

12-5698051 

5-4061202 

•006329114 

159 

25281 

4019679 

12-6095202 

5-4175015 

•006289308 

160 

25600 

4096000 

12-6491106 

5-4288352 

•006250000 

161 

25921 

4173281 

12-6885775 

5-4401218 

•006211180 

162 

26244 

4251528 

12-7279221 

5-4513618 

•006172840 

163 

26569 

4330747 

12-7671453 

5-4625556 

•006134969 

164 

26896 

4410944 

12-8062485 

5-4737037 

•006097561 

165 

27225 

4492125 

12-8452326 

5-4848066 

•006060606 

166 

27556 

4574296 

12-8840987 

5-4958647 

•006024096 

167 

27889 

4657463 

12-9228480 

5-5068784 

•005988024 

168 

28224 

4741632 

12-9614814 

5-5178484 

•005952381 

169 

28561 

4S26809 

13-0000000 

5-5287748 

•005917160 

170 

28900 

4913000 

13-0384048 

5-5396583 

•005882353 

171 

29241 

5000211 

13-0766968 

5-5504991 

•005847953 

172 

29584 

5088448 

13-1148770 

5-5612978 

•005813953 

173 

29929 

5177717 

13-1529464 

5-5720546 

•005780347 

174 

30276 

5268024 

13-1909060 

5-5827702 

•005747126 

175 

30625 

5359375 

13-2287566 

5-5934447 

•005714286 

176 

30976 

5451776 

13-2664992 

5-6040787 

•005681818 

177 

31329 

5545233 

13-3041347 

5-6146724 

•005649718 

178 

31684 

5639752 

13-3416641 

5-6252263 

•005617978 

179 

32041 

5735339 

13-3790882 

5-6357408 

•005586592 

180 

32400 

5832000 

13-4164079 

5-6462162 

•005555556 

181 

32761 

5929741 

13-4536240 

5-6566528 

•005524862 

182 

33124 

6028568 

13-4907376 

5-6670511 

•005494505 

183 

33489 

6128487 

13-5277493 

5-6774114 

•005464481 

184 

33856 

6229504 

13-5646600 

5-6877340 

•005434783 

185 

34225 

6331625 

13-6014705 

5-6980192 

•005405405 

186 

34596 

6434856 

13-6381817 

5-7082675 

•005376344 

187 

34969 

6539203 

13-6747943 

5-7184791 

•005347594 

188 

35344 

6644672 

13-7113092 

5-7286543 

•005319149 

189 

35721 

6751269 

13-7477271 

5-7387936 

•005291005 

190 

36100 

6859000 

13-7840488 

5-7488971 

•005263158 

191 

36481 

6987871 

13-8202750 

5-75S9652 

•005235602 

192 

36864 

7077888 

13-8564065 

5-7689982 

•005208333 

193 

37249 

7189517 

13-8924400 

5-7789966 

•005181347 

194 

37636 

7301384 

13-9283883 

5-7889604 

•005154639 

195 

38025 

7414875 

13-9642400 

5-7988900 

•005128205 

196 

38416 

7529536 

14-0000000 

5-8087857 

•005102041 

197 

38809 

7645373 

14-0356688 

5-81S6479 

•005076142 

198 

39204 

7762392 

14-0712473 

5-8284867 

•005050505 

199 

39601 

7880599 

14-1067360 

5-8382725 

•005025126 

200 

40000 

8000000 

14-1421356 

5-8480355 

•005000000 

201 

40401 

8120601 

14-1774469 

5-8577660 

•004975124 

202 

40804 

8242408 

14-2126704 

5-8674673 

•004950495 

203 

41209 

8365427 

14-2478068 

5-8771307 

•004926108 

204 

41616 

8489664 

14-2828569 

5-8867653 

•004901961 

205 

42025 

8615125 

14-3178211 

5-8963685 

•004878049 

206 

42436 

8741816 

14-3527001 

5-9059406 

•004854369 

207 

42849 

8869743 

14-3874946 

5-9154817 

•004830918 

208 

43264 

8998912 

14-4222051 

5-9249921 

•004807692 


















86 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares 

Cubes. 

\/ Roots. 

yj Roots. 

Reciprocals. 

209 

43681 

9129329 

14-4568323 

5-9344721 

•004784689 

210 

44100 

9261000 

14-4913767 

5-9439220 

•004761905 

211 

44521 

9393931 

14-5258390 

5-9533418 

•004739336 

212 

44944 

9528128 

14-5602198 

5-9627320 

•004716981 

213 

45369 

9663597 

14-5945195 

5-9720926 

•004694836 

214 

45796 

9800344 

14-6287388 

5-9814240 

•004672897 

215 

46225 

9938375 

14-6628783 

5-9907264 

•004651163 

216 

46656 

10077696 

14-6969385 

6-0000000 

•004629630 

217 

47089 

10218313 

14-7309199 

6-0092450 

•004608295 

218 

47524 

10360232 

14-7648231 

6-0184617 

•004587156 

219 

47961 

10503459 

14-79S6486 

6-0276502 

•004566210 

220 

48400 

10648000 

14-8323970 

6-0368107 

•004545455 

221 

48841 

10793861 

14-8660687 

6-0459435 

•004524887 

222 

. 49284 

10941048 

14-8996644 

6-0550489 

•004504505 

223 

49729 

11089567 

14-9331845 

6-0641270 

•004484305 

224 

50176 

11239424 

14-9666295 

6-0731779 

•004464286 

225 

50625 

11390625 

15-0000000 

6-0824020 

•004444444 

226 

51076 

11543176 

15-0332964 

6-0991994 

•004424779 

227 

51529 

11697083 

15-0665192 

6-1001702 

•004405286 

228 

51984 

11852352 

15-0996689 

6-1091147 

•004385965 

229 

52441 

12008989 

15-1327460 

6-1180332 

•004366812 

230 

52900 

12167000 

15-1657509 

6-1269257 

•004347826 

231 

53361 

12326391 

15-1986842 

6-1357924 

•004329004 

232 

53824 

12487168 

15-2315462 

6-1446337 

•004310345 

233 

54289 

12649337 

15-2643375 

6-1534495 

•004291845 

234 

54756 

12812904 

15-2970585 

6-1622401 

•004273504 

235 

55225 

12977875 

15-3297097 

6-1710058 

•004255319 

236 

55696 

13144256 

15-3622915 

6-1797466 

•004237288 

237 

56169 

13312053 

15-3948043 

6-1884628 

•004219409 

238 

56644 

13481272 

15-4272486 

6-1971544 

•004201681 

239 

57121 

13651919 

15-4596248 

6-2058218 

•004184100 

240 

57600 

13824000 

15-4919334 

6-2144650 

•004166667 

241 

58081 

13997521 

15-5241747 

6-2230843 

•004149378 

242 

58564 

14172488 

15-5563492 

6-2316797 

•004132231 

243 

59049 

14348907 

15-5884573 

6-2402515 

•004115226 

244 

59536 

14526784 

15-6204994 

6-2487998 

•004098361 

245 

60025 

14706125 

15-6524758 

6-2573248 

•004081633 

246 

60516 

14886936 

15-6843871 

6-26582 66 

•004065041 

247 

61009 

15069223 

15-7162336 

6-2743054 

•004048583 

248 

61504 

15252992 

15-7480157 

6-2827613 

•004032258 

249 

62001 

15438249 

15-7797338 

6-2911946 

•004016064 

250 

62500 

15625000 

15-8113883 

6-2996053 

•004000000 

251 

63001 

15813251 

15-8429795 

6-3079935 

•003984064 

252 

63504 

16003008 

15-8745079 

6-3163596 

•003968254 

253 

64009 

16194277 

15-9059737 

6-3247035 

•003952569 

254 

64516 

16387064 

15-9373775 

6-3330256 

•003937008 

255 

65025 

16581375 

15-9687194 

6-3413257 

•003921569 

256 

65536 

16777216 

16-0000000 

6-3496042 

•003906250 

257 

66049 

16974593 

16-0312195 

6-3578611 

•003891051 

258 

66564 

17173512 

16-0623784 

6-3660968 

•003875969 

259 

67081 

17373979 

16-0934769 

6-3743111 

•003861004 

260 

676*00, 

17576000 

16-1245155 

6-3825043 

•003846154 
























i able of Squares. Cubes, Square and Cube Roots. 


87 








N umber. 

Squares. 

Cubes, 

\/ Roots. 

\/ Roots. 

Reciprocals. 

261 

68121 

17779581 

16*1554944 

6*3906765 

*003831418 

282 

68644 

17984728 

16*1864141 

6*3988279 

*003816794 

263 

69169 

18191447 

16*2172747 

6*4069585 

•003802281 

264 

69696 

18399744 

16*2480768 

6*4150687 

*003787879 

265 

70225 

18609625 

16*2788206 

6*4231583 

•003773585 

266 

70756 

18821096 

16*3095064 

6*4312276 

*003759398 

267 

71289 

19034163 

16*3401346 

6*4392767 

*003745318 

268 

71824 

19248832 

16*3707055 

6*4473057 

*003731343 

269 

72361 

19465109 

16*4012195 

6*4553148 

*003717472 

270 

72900 

19683000 

16.4316767 

6*4633041 

*003703704 

271 

73441 

19902511 

16*4620776 

6*4712736 

*003690037 

272 

73984 

20123643 

16*4924225 

6*4792236 

*003676471 

273 

74529 

20346417 

16*5227116 

6*4871541 

*003663004 

274 

75076 

20570824 

16*5529454 

6*4950653 

•003649635 

275 

75625 

20796875 

16*5831240 

6*5029572 

*003636364 

276 

76176 

21024576 

16*6132477 

6*5108300 

*003623188 

277 

76729 

21253933 

16*6433170 

6-5186839 

*003610108 

278 

77284 

21484952 

16*6783320 

6-5265189 

*003597122 

279 

77841 

2 1 717639 

16*7032931 

6-5343351 

•003584229 

280 

78400 

21952000 

16*7332005 

6-5421326 

•003571429 

281 

78961 

22188041 

16*7630546 

6-5499116 

•003558719 

282 

79524 

22425768 

16*7928556 

6-5576722 

*003546099 

283 

800S9 

22665187 

16*8226038 

6-5654144 

•003533569 

284 

80656 

22906304 

16*8522995 

6-5731385 

•003521127 

285 

81225 

23149125 

16*8819430 

6-5808443 

*003508772 

286 

81796 

23393656 

16*9115345 

6-5885323 

•003496503 

287 

82369 

23639903 

16*9410743 

6-5962023 

•003484321 

288 

82944 

23887872 

16*9705627 

6.-6038545 

•003472222 

289 

83521 

24137569 

17*0000000 

6-6114890 

•003460208 

290 

84100 

24389000 

17*0293864 

6-6191060 

•003448276 

291 

84681 

24642171 

17*0587221 

6-6267054 

•003436426 

292 

85264 

24897088 

17*0880075 

6-6342874 

•003424658 

293 

85849 

25153757 

17*1172428 

6-6418522 

•003412969 

294 

86436 

25412184 

17*1464282 

6-6493998 

•003401361 

295 

87025 

25672375 

17*1755640 

6-6569302 

•003389831 

296 

87616 

25934836 

17*2046505 

6-6644437 

•003378378 

297 

88209 

26198073 

17*2336879 

6*6719403 

•003367003 

298 

88804 

26463592 

17*2626765 

6.6794200 

•003355705 

299 

89401 

26730899 

17*2916165 

6.6868831 

•003344482 

300 

90000 

27000000 

17*3205081 

6.6943295 

*003333333 

301 

90601 

27270901 

17*3493516 

6.7017593 

•003322259 

302 

91204 

27543608 

17*3781472 

6*7091729 

•003311258 

303 

91S09 

27818127 

17*4068952 

6-7165700 

•003301330 

304 

92416 

28094464 

17*4355958 

6*7239508 

•003289474 

305 

93025 

28372625 

17*4642492 

6*7313155 

•003278689 

306 

93636 

28652616 

17*4928557 

6*7386641 

•003267974 

307 

94249 

28934443 

17*5214155 

6-7459967 

•003257329 

308 

94864 

29218112 

17*5499288 

6-7533134 

•003246753 

309 

95481 

29503609 

17*5783958 

6*7606143 

•003236246 

310 

96100 

29791000 

17*6068169 

6*7678995 

•003225806 

311 

96721 

300S0231 

17*6351921 

6*7751690 

•003215434 

312 

97344 

30371328 

17*6635217 

6*7824229 

•003205128 
















Table of Squares, Cxtbes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

y/ Roots. 

3 / -- 

V Boots. 

--- ! 

Reciprocals. 

313 

97969 

30664297 

17-6918060 

6-7896613 

•003194888 

314 

98596 

30959144 

17-7200451 

6-7968844 

•003184713 

315 

99225 

31255875 

17-7482393 

6-8040921 

•003174603 

316 

99856 

31554496 

17*7763888 

6-8112847 

•003164557 

317 

100489 

31855013 

17-8044938 

6-8184620 

.003154574 

318 

101124 

32157432 

17-8325545 

6-8256242 

•003144654 

319 

101761 

32461759 

17-8605711 

6-8327714 

•003134796 

320 

102400 

32768000 

17-8885438 

6-8399037 

•003125000 

321 

103041 

33076161 

17-9164729 

6-8470213 

■003115265 

322 

103684 

33386248 

17-9443584 

6-8541240 

•003105590 

323 

104329 

33698267 

17-9722008 

6-8612120 

•003095975 

324 

104976 

34012224 

18-0000000 

6-8682855 

•003086420 

325 

105625 

34328125 

18-0277564 

6-8753433 

•003076923 

326 

106276' 

34645976 

18-0554701 

6-8823888 

•003067485 

327 

106929 

34965783 

18-0831413 

6-8894188 

•003048104 

328 

107584 

35287552 

18-1107703 

6-8964345 

•0030487S0 

329 

108241 

35611289 

18-1383571 

6-9034359 

•003039514 

330 

108900 

35937000 

18-1659021 

6-9104232 

•003030303 

331 

109561 

36264691 

18-1934054 

6-9173964 

•003021148 

332 

110224 

36594368 

18-2208672 

6-9243556 

•003.012048 

333 

110889 

36926037 

18-2482876 

6-9313088 

•003003003 

334 

111556 

37259704 

18-2756659 

6-9382321 

•002994012 

335 

112225 

37595375 

18-3030052 

6-9451496 

•002985075 

336 

112896 

37933056 

18-3303028 

6-9520533 

•002976190 

337 

113569 

38272753 

18-3575598 

6-9589434 

•002967359 

338 

114244 

38614472 

18-3847763 

6-9658198 

•002958580 

339 

114921 

38958219 

18-4119526 

6-9726826 

•002949853 

340 

115600 

39304000 

18-4390889 

6-9795321 

•002941176 

341 

116281 

39651821 

18-4661853 

6-9863681 

•002932551 

342 

116964 

40001688 

18-4932420 

6-9931906 

•002923977 

343 

117649 

40353607 

18-5202592 

7-0000000 

•002915452 

344 

118336 

40707584 

18-5472370 

7-0067962 

•002906977 

345 

119025 

41063625 

18-5741756 

7-0135791 

•002898551 

346 

119716 

41421736 

18-6010752 

7-0203490 

•002890173 

347 

120409 

41781923 

18-6279360 

7-0271058 

•002881844 

348 

121104 

42144192 

18-6547581 

7-0338497 

•002873563 

349 

121801 

42508549 

18-6815417 

7-0405860 

•002865330 

350 

122500 

42875000 

18-7082869 

7-0472987 

•002857143 

351 

123201 

43243551 

18-7349940 

7-0540041 

•002849003 

352 

123904 

43614208 

18-7616630 

7-0606967 

•002840909 

353 

124609 

43986977 

18*7882942 

7-0673767 

•002832861 

354 

125316 

44361864 

18-8148877 

7-0740440 

•002824859 

355 

126025 

44738875 

18-8414437 

7-0806988 

•002816901 

356 

126736 

45118016 

18-8679623 

7-0873411 

•002808989 

357 

127449 

45499293 

18-8944436 

7-0939709 

•002801120 

358 

128164 

45882712 

18-9208879 

7-1005885 

•002793296 

359 

128881 

46268279 

18-9472953 

7-1071937 

•002785515 

360 

129600 

4665600.0 

18-9736660 

7-1137866 

•002777778 

361 

130321 

47045831 

19-0000000 

7-1203674 

•002770083 

362 

131044 

47437928 

19-0262976 

7-1269360 

•002762431 

363 

131769 

47832147 

19-0525589 

7-1334925 

•002754821 

364 

132496 

48228544 

19-0787840 

7-1400370 

•002747253 

_i 





















Tafxe of Square, Cubes, Square and Cube Hoots. 


Number. 

Squares: 

Cubes. 

V Roots. 

$/ Roots. 

Reciprocals. 

365 

133225 

48627125 

19-1049732 

7-1465695 

*002739726 

366 

133956 

49027896 

19-1311265 

7-1530901 

•002732240 

367 

134689 

49430863 

19-1572441 

7-1595988 

•002724796 

368 

135424 

49836032 

19-1833261 

7-1660957 

•002717391 

369 

136161 

50243409 

19-2093727 

7-1725809 

•002710027 

370 

136900 

50653000 

19-2353841 

7-1790544 

•002702703 

371 

137641 

51064811 

19-2613603 

7-1855162 

•002695418 

372 

138384 

51478848 

19-2873015 

7-1919663 

•002688172 

373 

139129 

5189*5117 

19-3132079 

7-1984050 

•002680965 

374 

139876 

52313624 

19-3390796 

7-2048322 

•002673797 

375 

140625 

52734375 

19-3649167 

7-2112479 

•002666667 

376 

141376 

53157376 

19-3907194 

7-2176522 

•002659574 

377 

142129 

53582633 

19-4164878 

7-2240450 

•002652520 

378 

142884 

54010152 

19-4422221 

7-2304268 

•002645503 

379 

143641 

54439939 

19-4679223 

7-2367972 

•002638521 

380 

144400 

54872000 

19-4935887 

7-2431565 

•002631579 

381 

145161 

55306341 

19-5192213 

7-2495045 

•002624672 

382 

145924 

55742968 

19-5448203 

7-2558415 

•002617801 

383 

146689 

56181887 

19-5703858 

7-2621675 

•002610966 

384 

147456 

56623104 

19-5959179 

7-2684824 

•002604167 

385 

148225 

57066625 

19-6214169 

7-2747864 

•002597403 

386 

148996 

57512456 

19-6468827 

7-2810794 

•002590674 

387 

149769 

57960603 

19-6723156 

7-2873617 

•002583979 

388 • 

150544 

58411072 

19-6977156 

7-2936330 

•002577320 

389 

151321 

58863869 

19-7230829 

7-2998936 

•002570694 

390 

152100 

59319000 

19-7484177 

7-3061436 

•002564103 

391 

152881 

59776471 

19-7737199 

7-3123828 

•002557545 

392 

153664 

60236288 

19-7989899 

7-3186114 

•002551020 

393 

154449 

60698457 

19-8242276 

7-3248295 

•002544529 

394 

155236 

61162984 

19-8494332 

7*3310369 

•002538071 

395 

156025 

61629875 

19-8746069 

7-3372339 

•002531646 

396 

156816 

62099136 

19-8997487 

7-3434205 

•002525253 

397 

157609 

62570773 

19-9248588 

7-3495966 

•002518892 

398 

158404 

63044792 

19-9499373 

7-3557624 

•002512563 

399 

159201 

63521199 

19-9749844 

7-3619178 

•002506266 

400 

160000 

64000000 

20-0000000 

7-3680630 

•002500000 

401 

160801 

64481201 

20-0249844 

7-3741979 

•002493766 

402 

161604 

64964808 

20-0499377 

7-3803227 

•002487562 

403 

162409 

65450827 

20-0748599 

7-3864373 

•002481390 

404 

163216 

65939264 

20-0997512 

7-3925418 

•002475248 

405 

164025 

66430125 

20-1246118 

7-3986363 

•002469136 

406 

164836 

66923416 

20-1494417 

7-4047206 

•002463054 

407 

165649 

67419143 

20-1742410 

7-4107950 

•002457002 

408 

166464 

67917312 

20-1990099 

7-4168595 

•002450980 

409 

167281 

68417929 

20-2237484 

7-4229142 

•002444988 

410 

168100 

68921000 

20-2484537 

7-4289589 

•002439024 

411 

168921 

69426531 

20-2731349 

7-4349938 

•002433090 

412 

169744 

69934528 

20-2977831 

7-4410189 

•002427184 

413 

170569 

70444997 

20-3224014 

7-4470343 

•002421308 

414 

171396 

70957944 

20-3469899 

7-4530399 

•002415459 

415 

172225 

71473375 

20-3715488 

7-4590359 

•002409639 

416 

173056 

73991296 

20-3960781 

7-4650223 

•002406846, 

] 


.1 

























90 Table of Squares, Cubes, Square and Cube Roots. 


N umber. 

Squares. 

Cubes. 

Roots. 

\/ Roots. 

Reciprocals. 

417 

173889 

72511713 

20-4205779 

7-4709991 

•002398082 

418 

174724 

73034632 

20-4450483 

7-4769664 

•002392344 

419 

175561 

73560059 

20-4694895 

7-4829242 

•002386635 

420 

176400 

74088000 

20-4939015 

7-4888724 

•002380952 

421 

177241 

74618461 

20-5182845 

7-4948113 

•002375297 

422 

178084 

75151448 

20-5426386 

7-5007406 

•002369668 

423 

178929 

75686967 

20-5669638 

7-5066607 

•002364066 

424 

179776 

76225024 

20-5912603 

7-5125715 

•00235S491 

425 

180625 

76765625 

20-6155281 

7-5184730 

•002352941 

426 

181476 

77308776 

20-6397674 

7-5243652 

•002347418 

427 

182329 

77854483 

20-6639783 

7-5302482 

•002341920 

428 

183184 

78402752 

20-6881609 

7*5361221 

•002336449 

429 

184041 

78953589 

20-7123152 

7-5419867 

•002331002 

430 

184900 

79507000 

20-7364414 

7*5478423 

•002325581 

431 

185761 

80062991 

20-7605395 

7-5536888 

•002320186 

432 

186624 

80621568 

20-7846097 

7*5595263 

•002314815 

433 

187489 

81182737 

20-8086520 

7-5653548 

•002309469 

434 

188356 

81746504 

20-8326667 

7-5711743 

•002304147 

435 

189225 

82312875 

20-8566536 

7-5769849 

•002298851 

436 

190096 

82881856 

20-8806130 

7-5827865 

•002293578 

437 

190969 

83453453 

20-9045450 

7-5885793 

•002288330 

438 

191844 

84027672 

20-9284495 

7-5943633 

•002283105 

439 

192721 

84604519 

20-9523268 

7-6001385 

•002277904 

440 

193600 

85184000 

20-9761770 

7-6059049 

•002272727 

441 

194481 

85766121 

21-0000000 

7-6116626 

•002267574 

442 

195364 

86350888 

21-0237960 

7-6174116 

•002262443 

443 

196249 

86938307 

21-0475652 

7-6231519 

•002257336 

444 

197136 

87528384 

21-0713075 

7-6288837 

•002252252 

445 

198025 

88121125 

21-0950231 

7-6346067 

•002247191 

446 

198916 

88716536 

21-1187121 

7-6403213 

•002242152 

447 

199809 

89314623 

21-1423745 

7-6460272 

•002237136 

448 

200704 

89915392 

21-1660105 

7-6517247 

•002232143 

449 

201601 

90518849 

21-1896201 

7*6574138 

•002227171 

450 

202500 

91125000 

21-2132034 

7-6630943 

•002222222 

451 

203401 

91733851 

21-2367606 

7-6687665 

•002217295 

452 

204304 

92345408 

21-2602916 

7-6744303 

•002212389 

453 

205209 

92959677 

21-2837967 

7-6800857 

•002207506 

454 

206116 

93576664 

21-3072758 

7-6857328 

*002202643 

455 

207025 

94196375 

21-3307290 

7-6913717 

•0021 7S02 

456 

207936 

94818816 

21-3541565 

7-6970023 

•002192982 

457 

208849 

95443993 

21-3775583 

7-7026246 

•002188184 

458 

209764 

96071912 

21-4009346 

7-7082388 

•002183406 

459 

210681 

96702579 

21-4242853 

7*7188448 

•002178649 

460 

211600 

97336000 

21-4476106 

7-7194426 

•002173913 

461 

212521 

97972181 

21-4709106 

7-7250325 

•002169197 

462 

213444 

98611128 

21-4941853 

7-7306141 

•002164502 

463 

214369 

99252847 

21-5174348 

7-7361877 

•002159827 

464 

215296 

99897344 

21-5406592 

7-7417532 

•002155172 

465 

216225 

100544625 

21-5638587 

7-7473109 

•002150538 

468 

217156 

101194696 

21-5870331 

7-7528606 

•002145923 

467 

218089 

101847563 

21-6101828 

7-7584023 

•002141328 

468 

219024 

102503232 

21-6333077 1 

7-7639361 

•002136752 

















Table oh Squares, Cubes, Square and Cube Roots. 


N umber. 

Squares. 

Cubes. 

\/ Roots. 

4 / Roots. 

Reciprocals. 

469 

219961 

103161709 

21-6564078 

7-7694620 

•002132196 

470 

220900 

103823000 

21-6794834 

7-7749801 

•002127660 

471 

221841 

104487111 

21-7025344 

7-7804904 

•002123142 

472 

222784 

105154048 

21-7255610 

7-7859928 

•002118644 

473 

223729 

105828817 

21-7485632 

7-7914875 

•002114165 

474 

224676 

106496424 

21-7715411 

7-7969745 

•002109705 

475 

225625 

107171875 

21-7944947 

7-8024538 

•002105263 

.478 

226576 

107850176 

21-8174242 

7-8079254 

•002100840 

477 

227529 

108531333 

21-8403297 

7-8133892 

•002096436 

478 

228484 

109215352 

21-8632111 

7-S188456 

•002092050 

479 

229441 

109902239 

21-8860686 

7-8242942 

•002087683 

480 

230400 

110592000 

21-9089023 

7-8297353 

•002083333 

481 

231361 

111284641 

21-9317122 

7-8351688 

•002079002 

482 

232324 

111980168 

21-9544984 

7-8405949 

•002074689 

483 

233289 

112678587 

21-9772610 

7-8460134 

•002070393 

484 

234256 

113379904 

22-0000000 

7-8514244 

•002066116 

485 

235225 

114084125 

22-0227155 

7-8568281 

•002061856 

486 

236196 

114791256 

22-0454077 

7-8622242 

•002057613 

487 

237169 

115501303 

22-0680765 

7-8676130 

•002053388 

488 

238144 

116214272 

22-0907220 

7-8729944 

•002049180 

489 

239121 

116930169 

22-1133444 

7-8783684 

•002044990 

490 

240100 

117649000 

22-1359436 

7-8837352 

•002040816 

491 

241081 

118370771 

22-1585198 

7-8890946 

•002036660 

492 

242064 

119095488 

22-1810730 

7-8944468 

•002032520 

493 

243049 

119823157 

22-2036033 

7-8997917 

•002028398 

494 

244036 

120553784 

22-2261108 

7-9051294 

•002024291 

495 

245025 

121287375 

22-2485955 

7-9104599 

•002020202 

496 

246016 

122023936 

22-2710575 

7-9157832 

•002016129 

497 

247009 

122763473 

22-2934968 

7-9210994 

•002012072 

498 

248004 

123505992 

22-3159136 

7-9264085 

•002008032 

499 

249001 

124251499 

22-3383079 

7-9317104 

•002004008 

500 

250000 

125000000 

22-3606798 

7-9370053 

•002000000 

501 

251001 

125751501 

22-3830293 

7-9422931 

•001996008 

502 

252004 

126506008 

22-4053565 

7-9475739 

•001992032 

503 

253009 

127263527 

22-4276615 

7-9528477 

•001988072 

504 

254016 

128024064 

22-4499443 

7*9581144 

•001984127 

505 

255025 

128787625 

22-4722051 

7-9633743 

•001980198 

506 

256036 

129554216 

22-4944438 

7-9686271 

•001976285 

507 

257049 

130323843 

22-5166605 

7-9738731 

•001972387 

508 

258064 

131096512 

22-5388553 

7-9791122 

•001968504 

509 

259081 

131*72229 

22-5610283 

7-9843444 

•001964637 

510 

260100 

132651000 

22-5831796 

7-9895697 

•001960784 

511 

261121 

133432831 

22-6053091 

7-9947883 

•001956947 

512 

262144 

134217728 

22-6274170 

8-0000000 

•001953125 

513 

263169 

135005697 

22-6495033 

8-0052049 

•001949318 

514 

264196 : 

135796744 

22-6715681 

8-0104032 

•001945525 

515 

265225 

136590875 

22-6936114 

8-0155946 

•001941748 

516 

266256 

137388096 

22-7156334 

8-0207794 

•001937984 

517 

267289 

138188413 

22-7376341 

8-0259574 

•001934236 

518 

268324 

138991832 

22-7596134 

8-0311287 

•001930502 

519 

269361 

139798359 

22-7815715 

8-0362935 

•001926782 

520 

270400 1 

140608000 ! 

22-8035085 1 

8-0414515 

•001923077 
























D2 Table of Squares, Cures, Square and Cube Hoots. 


Number, 

Squares. 

Cubes. 

\/ Roots. 

.. /- 

</ Roots. 

Reciprocals. 

521 

271441 

141420761 

22-8254244 

8-0466030 

•001919386 

522 

272484 

142236648 

22-8473193 

8-0517479 

•001915709 

523 

273529 

143055667 

22-8691933 

8-0568862 

•001912046 

524 

274576 

143877824 

22*8910463 

8-0620180 

•001908397 

525 

275625 

144703125 

22-9128785 

8-0671432 

•001904762 

526 

276676 

145531576 

22-9346899 

8-0722620 

•001901141 

527 

277729 

146363183 

22-9564806 

8-0773743 

•001897533 

528 

278784 

147197952 

22-9782506 

8-0824800 

•001S93939 

529 

279841 

148035889 

23-0000000 

8-0875794 

•001890359 

530 

280900 

148877001 

23-0217289 

8-0926723 

' *001886792 

531 

281961 

149721291 

23-0434372 

8-0977589 

•001883239 

532 

283024 

150568768 

23-0651252 

8-1028390 

•001879699 

533 

284089 

151419437 

23-0867928 

8-1079128 

•001876173 

534 

285156 

152273304 

23-1084400 

8-1129803 

•001872659 

535 

286225 

153130375 

23-1300670 

8-1180414 

•001869159 

536 

287296 

153990656 

23-1516738 

8-1230962 

•001865672 

537 

288369 

154854153 

23-1732605 

8-1281447 

•001862197 

538 

289444 

155720872 

23-1948270 

8-1331870 

•001858736 

539 

290521 

156590819 

23-2163735 

8-1382230 

•0018552S8 

540 

291600 

157464000 

23-2379001 

8-1432529 

•001851852 

541 

292681 

J58340421 

23-2594067 

8-1482765 

•001848429 

542 

293764 

159220088 

23-2808935 

8-1532939 

•001845018 

543 

294849 

160103007 

23-3023604 

8-1583051 

•001841621 

544 

295936 

160989184 

23-3238076 

8-1633102 

•001838235 

545 

297025 

161878625 

23-3452351 

8-1683092 

•001834862 

546 

298116 

162771336 

23-3666429 

8-1733020 

•001831502 

547 

299209 

163667323 

23-3880311 

8-1782888 

•001828154 

548 

300304 

164566592 

23-4093998 

8-1832695 

•001824818 

549 

301401 

165469149 

23-4307490 

8-1882441 

•001821494 

550 

302500 

166375000 

23-4520788 

8-1932127 

•001818182 

551 

303601 

167284151 

23-4733892 

8-1981753 

•001814882 

552 

304704 

168196608 

23-4946802 

8-2031319 

•001811594 

553 

305809 

169112377 

23-5159520 

8-2080825 

•001808318 

554 

306916 

170031464 

23-5372046 

8-2130271 

•001805054 

555 

308025 

170953875 

23-5584380 

8-2179657 

•001801802 

556 

309136 

171879616 

23-5796522 

8-2228985 

•001798561 

557 

310249 

172808693 

23-6008474 

8-2278254 

•001795332 

558 

311364 

173741112 

23-6220236 

8-2327463 

•001792115 

559 

312481 

174676879 

23-6431808 

8-2376614 

•001788909 

560 

313600 

175616000 

23-6643191 

8-2425706 

•001785714 

561 

314721 

176558481 

23-6854386 

8-2474740 

•001782531 

562 

315844 

177504328 

23-7065392 

8-2523715 

•001779359 

563 

316969 

178453547 

23-7276210 

8-2572635 

•001776199 

564 

318096 

179406144 

23-7486842 

8-2621492 

•001773050 

565 

319225 

180362125 

23-7697286 

8-2670294 

•001769912 

566 

320356 

181321496 

23-7907545 

8-2719039 

•001766784 

567 

3214S9 

182284263 

23-8117618 

8-2767726 

•001763668 

568 

322624 

183250432 

23-8327506 

8-2816255 

•001760503 

569 

323761 

184220009 

23-8537209 

8-2864928 

•001757469 

570 

324900 

185193000 

23-8746728 

8-2913444 

•001754386 

571 

326041 

186169411 

23-8956063 

8-2961903 

•001751313 

572 

327184 

187149248 

23-9165215 

8-3010304 

•001748252 


















Table of Squares, Cubes, Square and Cube Roots. 


93 


Number. 

Squares. 

Cubes. 

Roots. 

V Hoots. 

Reciprocals. 

573 

328329 

188132517 

23-9374184 

8-3058651 

•001745201 

574 

329476 

189119224 

23-9582971 

8-3106941 

•001742160 

575 

330625 

190109375 

23-9791576 

8-3155175 

•001739130 

576 

331776 

191102976 

24-0000000 

8-3203353 

•001736111 

577 

332927 

192100033 

24-0208243 

8-3251475 

•001733102 

578 

334084 

193100552 

24-0416306 

8-3299542 

•001730104 

579 

335241 

194104539 

24-0624188 

8-3347553 

•001727116 

580 

336400 

195112000 

24-0831891 

8-3395509 

•001724138 

581 

337561 

196122941 

24-1039416 

8-3443410 

•001721170 

582 

338724 

197137368 

24-1246762 

8-3491256 

•001718213 

583 

339889 

198155287 

24-1453929 

8-3539047 

•001715266 

584 

341056 

.199176704 

24-1660919 

8-3586784 

•001712329 

585 

342225 

200201625 

24-1867732 

8-3634466 

•001709402 

586 

343396 

201230056 

24-2074369 

S-36S2095 

•001706485 

587 

344569 

202262003 

24-2280829 

8-3729668 

•001703578 

588 

345744 

203297472 

24-2487113 

8-3777188 

•001700680 

589 

346921 

204336469 

24-2693222 

8-3824653 

•001697793 

590 

348100 

205379000 

24-2899156 

8-3872065 

•001694915 

591 

349281 

206425071 

24-3104996 

8-3919428 

•001692047 

592 

350464 

207474688 

24-3310501 

8-3966729 

•001689189 

593 

351649 

208527857 

24-3515913 

8-4013981 

•001686341 

594 

352836 

209584584 

24-3721152 

8-4061180 

•001683502 

595 

354025 

210644875 

24-3926218 

8-4108326 

■001680672 

596 

355216 

211708736 

24-4131112 

8-4155419 

•001677852 

597 

356409 

212776173 

24-4335834 

8-4202460 

•001675042 

598 

357604 

213847192 

24-4540385 

8-4249448 

•001672241 

599 

358801 

214921799 

24-4744765 

8-4296383 

•001669449 

600 

360000 

216000000 

24-4948974 

8-4343267 

•001666667 

601 

361201 

217081801 

24-5153013 

8-4390098 

•001663894 

602 

362404 

218167208 

24-5356883 

8-4436877 

•001661130 

603 

363609 

219256227 

24-5560583 

8-4483605 

•001658375 

604 

364816 

220348864 

24-5764115 

8-4530281 

•001655629 

605 

366025 

221445125 

24-5967478 

8-4576906 

•001652893 

606 

367236 

222545016 

24-6170673 

8-4623479 

•001650165 

607 

368449 

223648543 

24-6373700 

8-4670001 

•001647446 

608 

369664 

224755712 

24*657 6560 

8-4716471 

•001644737 

609 

370881 

225866529 

24-6779254 

8-4762892 

•001642036 

610 

372100 

226981000 

24-6981781 

8-4809261 

•001639344 

611 

373321 

228099131 

24-7184142 

8-4855579 

•001636661 

612 

374544 

229220928 

24-7386338 

8-4901848 

•001633987 

613 

375769 

230346397 

24-7588368 

8-4948065 

•001631321 

614 

376996 

231475544 

24-7790234 

8-4994233 

•001628664 

615 

378225 

232608375 

24-7991935 

8-5040350 

•001626016 

616 

379456 

233744896' 

24-8193473 

8-5086417 

•001623377 

617 

380689 

234885113 

24-8394847 

8-5132435 

•001620746 

618 

381924 

236029032 

24-8596058 

8-5178403 

•001618123 

619 

383161 

237176659 

24-8797106 

8-5224331 

•001615509 

620 

384400 

238328000i 

24-8997992 

8-5270189 

•001612903 

621 

385641 

239483061 

24-9198716 

8-5316009 

•001610306 

822 

386884 

2406418481 

24-9399278 

8-5361780 

•001607717 

623 

388129! 

241804367 

24-9599679 

8-5407501 

•001605136 

624 | 

389376! 

2429706241 

24-9799920 

8-5453173 

001602564 






















94 Tabus op Squares, Cubes, Square and Cube Roots. 


Number, i 

Squares. 1 

Cubes. 

\/ Roots. 

4/ Roots. 

Reciprocals. 

625 

390625 

244140625 

25-0000000 

8-5498797 

•001600000 

626 

391876 

245134376 

25-0199920 

8-5544372 

•601597444 

627 

393129 

246491883 

25-0399681 

8-5589899 

•001594896 

62S 

394384 

247673152 

25-0599282 

8-5635377 

•001592357 

629 

395641 

248858189 

25-0798724 

8-5680807 

•001589825 

630 

396900 

250047000 

25-0998008 

8-5726189 

•001587302 

631 

398161 

251239591 

25-1197134 

8-5771523 

•001584786 

632 

399424 

252435968 

25-1396102 

8-5816809 

•001582278 

633 

400689 

253636137 

25-1594913 

8-5862247 

•001579779 

634 

401956 

254840104 

25-1793566 

8-5907238 

•001577287 

635 

403225 

256047875 

25-1992063 

8-5952380 

•001574803 

636 

404496 

257259456 

25-2190404 

8-5997476 

•001572327 

637 

405769 

258474853 

25-2388589 

8-6042525 

•001569859 

638 

407044 

259694072 

25-2586619 

8-6087526 

•001567398 

639 

408321 

260917119 

25-2784493 

8-6132480 

•001564945 

640 

409600 

262144000 

25-2982213 

8-6177388 

•001562500 

641 

410881 

263374721 

25-3179778 

S-6222248 

•001560062 

642 

412164 

264609288 

25-3377189 

8-6267063 

•001557632 

643 

413449 

265847707 

25-3574447 

8-6311830 

•001555210 

644 

414736 

267089984 

25-3771551 

8-6356551 

•001552795 

645 

416025 

268336125 

25-3968502 

8-6401226 

•001550388 

646 

417316 

269585136 

25-41&5302 

8-6445855 

•001547988 

647 

418609 

270840023 

25-4361947 

8-6490437 

•001545595 

648 

419904 

272097792 

25-4558441 

8-6534974 

•001543210 

649 

424201 

273359449 

25-4754784 

8-6579465 

•001540832 

650 

422500 

274625000 

25-4950976 

8-6623911 

•001538462 

651 

423801 

275894451 

25-5147013 

8-6668310 

•001536098 

652 

425104 

277167808 

25-5342907 

8-6712665 

•001533742 

653 

426409 

278445077 

25-5538647 

8-6756974 

•001531394 

654 

427716 

279726264 

25-5734237 

8-6801237 

•001529052 

655 

429025 

281011375 

25-5929678 

8-6845456 

•001526718 

656 

430336 

282300416 

25-6124969 

8-6889630 

•001524390 

657 

4316 

283593393 

25-6320112 

8-6933759 

•001522070 

658 

432964 

284890312 

25-6515107 

8-6977843 

*001519757 

659 

434281 

286191179 

25-6709953 

8-7021882 

•001517451 

660 

435600 

287496000 

25-6904652 

8-7065877 

•001515152 

661 

436921 

28SS04781 

25.7099203 

8-7109827 

•001512859 

662 

438244 

290117528 

25-7293607 

8-7153734 

•001510574 

663 

439569 

291434247 

25-7487864 

8-7197596 

•00150S296 

664 

440896 

292754944 

25-7681975 

8-7241414 

*001506024 

665 

442225 

294079625 

25-7875939 

8-7285187 

•001503759 

666 

443556 

295408296 

25-8069758 

8-7328918 

*001501502 

667 

444S89 

296740963 

25-8263431 

8-7372604 

•001499250 

668 

446224 

298077632 

25-8456960 

8-7416246 

•001497006 

669 

447561 

299418309 

25-8650343 

8-7459846 

•001494768 

670 

448900 

300763000 

25-88435S2 

8-7503401 

•001492537 

671 

450241 

302111711 

25-9036677 

8-7546913 

•001490313 

672 

451584 

303464448 

25-9229628 

8-7590383 

•001488095 

673 

452929 

304821217 

25-9422435 

8-7633809 

•001485884 

674 

454276 

306182024 

25-9615100 

Q W/T1 AO 

u 1 Ul ( 

•001483680 

676 

455625 

307546875 

25-9807621 

8-7720532 

•001481481 

676 

456976 

308915776 

26-0000000 

8*7763830 

•001479290 


















Tabu: of Squares, Cubes, Square attc> Cube Roots. 


95 


Number. 

Squares. 

Cubes. 

a/ Roots. 

Roots, 

Reciprocals. 

677 

458329 

310288733 

26-0192237 

8-7807084 

•001477105 

67S 

459684 

311665752 

26-0384331 

8-7850296 

•001474926 

679 

461041 

313046839 

26-0576284 

8-7893466 

•001472754 

680 

462400 

314432000 

26-0768096 

8-7936593 

•001470588 

681 

463761 

315821241 

26-0959767 

8-7979679 

•001468429 

682 

465124 

317214568 

26-1151297 

8-8022721 

•001466276 

683 

466489 

318611987 

26-1342687 

8-8065722 

•001464129 

684 

467856 

320013504 

26-1533937 

8-8108681 

•001461988 

685 

469225 

321419125 

26-1725047 

8-8151598 

•001459854 

686 

470596 

322828856 

26-1916017 

8-8194474 

•001457726 

687 

471969 

324242703 

26-2106848 

8-8237307 

•001455604 

688 

473344 

325660672 

26-2297541 

8-8280099 

•001453488 

689 

474721 

327082769 

26-2488095 

8-8322850 

•001451379 

690 

476100 

328509000 

26-2678511 

8-8365559 

•001449275 

691 

477481 

329939371 

26-2868789 

8-8408227 

•001447178 

692 

478864 

331373888 

26-3058929 

8-8450854 

•001445087 

693 

480249 

332812557 

26-3248932 

8-8493440 

•001443001 

694 

481636 

334255384 

26-3438797 

8-8535985 

•001440922 

695 

483025 

335702375 

26-3628527 

8-8578489 

•001438849 

696 

484416 

337153536 

26-3818119 

8*8620952 

•001436782 

697 

485809 

338608873 

26-4007576 

8-8663375 

•001434720 

698 

487204 

340068392 

26-4196896 

8-8705757 

•001432665 

699 

488601 

341532099 

26-4386081 

8-8748099 

•001430615 

700 

490000 

343000000 

26-4575131 

8-8790400 

•001428571 

701 

491401 

344472101 

26-4764046 

8-8832661 

•001426534 

702 

492804 

345948408 

26-4952826 

8-8874882 

•001424501 

703 

494209 

347428927 

26-5141472 

8-8917063 

•001422475 

704 

495616 

348913664 

26-5329983 

8-8959204 

•001420455 

705 

497025 

350402625 

26-5518361 

8-9001304 

•001418440 

706 

498436 

351895S16 

26-5706605 

8-9043366 

•001416431 

707 

499849 

353393243 

26-5894716 

8-9085387 

•001414427 

708 

501264 

354894912 

26-6082694 

8-9127369 

•001412429 

709 

502681 

356400829 

26-6270539 

8-9169311 

•001410437 

710 

504100 

357911000 

26-6458252 

8-9211214 

•001408451 

711 

505521 

359425431 

26-6645833 

8-9253078 

•001406470 

712 

506944 

360944128 

26-6833281 

8-9294902 

•001404494 

713 

508369 

362467097 

26-7020598 

8-9336687 

•001402525 

714 

509796 

363994344 

26-7207784 

8-9378433 

•001400560 

715 

511225 

365525875 

26-7394839 

8-9420140 

•001398601 

716 

512656 

367061696 

26-75S1763 

8-9461809 

•001396648 

717 

514089 

368601813 

26-7768557 

8-9503438 

•001394700 

718 

515524 

370146232 

26-7955220 

8-9545029 

•001392758 

719 

516961 

371694959 

26-8141754 

8-9586581 

•001390821 

720 

518400 

373248000 

26-8328157 

8-9628095 

•001388889 

721 

519841 

374805361 

26-8514432 

8-9669570 

•001386963 

722 

521284 

376367048 

26-8700577 

8-9711007 

•001385042 

723 

522729 

377933067 

26-8886593 

8-9752406 

•001383126 

724 

524176 

379503424 

26-9072481 

. 8-9793766 

•001381215 

725 

525625 

381078125 

26-9258240 

8-9835089 

•001379310 

726 

527076 

382657176 

26-9443S72 

8-9876373 

•001377410 

727 

528529 

384240583 

26-9629375 

8-9917620 

•001375516' 

728 

529984 

385828352 

26-9814751 

8-9958899 

•001373626 

















Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

\/ Roots. 

V 7 Roots. 

Reciprocals. 

729 

531441 

887420489 

27-0000000 

9-0000000 

001371742 

730 

532900 

389017000 

27-0185122 

9-0041134 

•001369863 

731 

534361 

390617891 

27-0370117 

9-0082229 

•0013679S9 

732 

535824 

392223168 

27-0554985 

9-0123288 

•001366120 

733 

537289 

393832837 

27-0739727 

9-0164309 

•001364256 

734 

538756 

395446904 

27-0924344 

9-0205293 

•001362398 

735 

540225 

397065375 

27-1108834 

9-0246239 

•001360544 

736 

541696 

398688256 

27-1293199 

9-0287149 

•001358696 

737 

543169 

400315553 

27-1477149 

9-0328021 

•001356852 

738 

544644 

401947272 

27-1661554 

9-0368857 

•001355014 

739 

546121 

403583419 

27-1845544 

9-0409655 

•001353180 

740 

547600 

405224000 

27-2029140 

9-0450419 

•001351351 

741 

549081 

406869021 

27-2213152 

9-0491142 

•001349528 

742 

550564 

40851S488 

27-2396769 

9-0531831 

•001347709 

743 

552049 

410172407 

27-2580263 

9-0572482 

•001345895 

744 

553536 

411830784 

27-2763634 

9 0613098 

•001344086 

745 

555025 

413493625 

27-2946881 

9-0653677 

•001342282 

746 

556516 

415160936 

27-3130006 

9-0694220 

•001340483 

747 

558009 

416832723 

27-3313007 

9-0734726 

•001338688 

748 

559504 

418508992 

27-3495887 

9-0775197 

•001336898 

749 

561001 

420189749 

27-3678644 

9-0815631 

•001335113 

750 

562500 

421875000 

27-3861279 

9-0856030 

•001333333 

751 

564001 

423564751 

27-4043792 

9-0896352 

•001331558 

752 

565504 

425259008 

27-4226184 

9-0936719 

•001329787 

753 

567009 

426957777 

27-4408455 

9-0977010 

•001328021 

754 

568516 

428661064 

27-4590604 

9-1017265 

•001326260 

755 

570025 

430368875 

27-4772633 

9-1057485 

•001324503 

756 

571536 

432081216 

27-4954542 

9-1097669 

•001322751 

757 

573049 

43379S093 

27-5136330 

9-1137818 

•001321004 

758 

574564 

435519512 

27-5317998 

9-1177931 

•001319261 

759 

576081 

437245479 

27-5499546 

9-1218010 

•001317523 

760 

577600 

438976000 

27-5680975 

9-1258053 

•001315789 

761 

579121 

440711081 

27-5862284 

9-1298061 

•001314060 

762 

580644 

442450728 

27-6043475 

9-1338034 

•001312336 

763 

582169 

444194947 

27-6224546 

9-1377971 

•001310616 

764 

583696 

445943744 

27-6405499 

9-1417874 

•001308901 

765 

585225 

447697125 

27-6586334 

9-1457742 

•001307190 

766 

586756 

449455096 

27-6767050 

9-1497576 

•001305483 

767 

588289 

451217663 

27-6947648 

9-1537375 

•001303781 

768 

589824 

452984832 

27-7128129 

9-1577139 

•001302083 

769 

591361 

454756609 

27-7308492 

9-1616869 

•001300390 

770 

592900 

456533000 

27*7488739 

9-1656565 

•001298701 

771 

594441 

458314011 

27-7668868 

9-1696225 

•001297017 

772 

595984 

460099648 

27-7848880 

9-1735852 

•001295337 

773 

597529 

461889917 

27-8028775 

9-1775445 

•001293661 

774 

599076 

463684824 

27-8208555 

9-1815003 

•001291990 

775 

600625 

465484375 

27-8388218 

9-1854527 

•O01290323 

776 

602176 467288576 

27-8567766 

9-1894018 

•001288660 

777 

603729 

469097433 

27-8747197 

9-1933474 

-001287001 

778 

605284 

470910952 

27-8926514 

9-1972897 

•001285347 

779 

606841 

472729139 

27-9105715 

9-2012286 

•001283697 

780 

1 

6“«400 

474552000 

27-9284801 

9-2051641 

•001282051 
























1 'abj.e of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 





Cubes. 

V Roots. 

] v/ Routs. 

Reciprocals. 

781 

609961 

476379541 

27-9463772 

9-2090962 

•001280410 

782 

611524 

478211768 

27-9642629 

9-2130250 

•001278772 

783 

613089 

480048687 

27-9821372 

9-2169505 

•001277139 

784 

614656 

481890304 

28-0000000 

9-2208726 

•001275510 

785 

616225 

483736625 

28-0178515 

9-2247914 

•001273885 

786 

617796 

485587656 

28-0356915 

9-2287068 

•001272265 

787 

619369 

487443403 

28-0535203 

9-2326189 

•001270648 

788 

620944 

489303872 

28-0713377 

9-2365277 

•001269036 

789 

622521 

491169069 

28-0891438 

9-2404333 

•001267427 

790 

624100 

493039000 

28-1069386 

9-2443355 

•001265823 

791 

625681 

494913671 

28-1247222 

9-2482344 

•001264223 

792 

627264 

496793088 

28-1424946 

9-2521300 

•001262626 

793 

628849 

498677257 

28-1602557 

9-2560224 

•001261034 

794 

630436 

500566184 

28-1780056 

9-2599114 

•001259446 

795 

632025 

502459875 

28-1957444 

9-2637973 

•001257862 

796 

633616 

504358336 

28-2134720 

9-2676798 

•001256281 

797 

635209 

506261573 

28-2311884 

9-2715592 

•00 1 254705 

798 

636804 

508169592 

28-2488938 

9-2754352 

•001253133 

799 

638401 

510082399 

28-2665881 

9-2793081 

•001251564 

800 

640000 

512000000 

28-2842712 

9-2831777 

•001250000 

801 

641601 

513922401 

28-3019434 

9-2870444 

•001248439 

802 

643204 

515849608 

28-3196045 

9-2909072 

•001246883 

803 

644809 

517781627 

28-3372546 

9-2947671 

•001245330 

804 

646416 

519718464 

28-3548938 

9-2986239 

•001243781 

805 

648025 

521660125 

28-3725219 

9-3024775 

•001242236 

806 

649636 

523606616 

28-3901391 

9-3063278 

•001240695 

807 

651249 

525557943 

28-4077454 

9-3101750 

•001239157 

808 

652864 

527514112 

28-4253408 

9-3140190 

•001237624 

809 

654481 

529475129 

28-4429253 

9-3178599 

•001236094 

810 

656100 

531441000 

28-4604989 

9-3216975 

•001234568 

811 

657721 

533411731 

28-4780617 

9-3255320 

■001233046 

812 

659344 

535387328 

28-4956137 

9-3293634 

•001231527 

813 

660969 

537367797 

28-5131549 

9-3331916 

•001230012 

814^ 

662596 

539353144 

28-5306852 

9-3370167 

•001228501 

815 

664225 

541343375 

28-5482048 

9-3408386 

•001226994 

816 

665856 

54333S496 

28-5657137 

9-3446575 

•001225499 

817 

667489 

545338513 

28-5832119 

9-3484731 

•001223990 

818 

669124 

547343432 

28-6006993 

9-3522857 

•001222494 

819 

670761 

549353259 

28-6181760 

9-3560952 

•001221001 

820 

672400 

551368000 

28-6356421 

9-3599016 

•001219512 

821 

674041 

553387661 

28-6530976 

9-3637049 

•001218027 

822 

675684 

555412248 

28-6705424 

9-3675051 

•001216545 

823 

677329 

557441767 

28-6879716 

9-3713022 

•001215067 

824 

678976 

559476224 

28-7054002 

9-3750963 

•001213592 

825 

680625 

561515625 

28-7228132 

9-3788873 

•001212121 

826 

682276 

563559976 

28-7402157 

9-3826752 

•001210654 

827 

683929 

505609283 

28-7576077 

9-3864600 

•001209190 

828 

685584 

567663552 

28-7749891 

9-3902419 

•001207729 

829 

687241 

569722789 

28-7923601 

9-3940206 

•001206273 

830 

688900 

571787000 

28-8097206 

9-3977964 

•001204819 

831 

690561 

573856191 

28-8270706 

9-4015691 

•001203369 

832 

692224 

575930368 

28-8444102 

9-4053387 

•001201923 


7 


























98 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

\/ Roots. 

V Roots. 

Reciprocals 

833 

693889 

578009537 

28-8617394 

9-4091054 

•001200480 

834 

695556 

580093704 

28*8790582 

9-4128690 

•001199041 

835 

697225 

582182875 

28-8963666 

9-4166297 

•001197605 

836 

698896 

584277056 

28-9136646 

9-4203873 

•001196172 

837 

700569 

586376253 

28-9309523 

9-4241420 

•001194743 

838 

702244 

588480472 

28-9482297 

9-4278936 

•001193317 

839 

703921 

590589719 

28-9654967 

9-4316423 

•001191895 

840 

705600 

592704000 

28-9827535 

9-4353800 

•001190476 

841 

707281 

594823321 

29-0000000 

9-4391307 

•001189061 

842 

708964 

596947688 

29-0172363 

9-4428704 

•001187648 

843 

710649 

599077107 

29-0344623 

9-4466072 

•001186240 

844 

712336 

601211584 

29-0516781 

9-4503410 

•001184834 

845 

714025 

603351125 

29-0688837 

9-4540719 

•001183432 

846 

715716 

605495736 

29-0860791 

9*4577999 

•001182033 

847 

717409 

607645423 

29-1032644 

9-4615249 

•001180638 

848 

719104 

609800192 

29-1204396 

9-4652470 

•001179245 

849 

720801 

611960049 

29-1376046 

9-4689661 

•001177856 

850 

722500 

614125000 

29-1547595 

9-4726824 

•001176471 

851 

724201 

616295051 

29-1719043 

9-4763957 

•001175088 

852 

725904 

618470208 

29-1890390 

9-4801061 

•001173709 

853 

727609 

620650477 

29-2061637 

9-4838136 

•001172333 

854 

729316 

622835864 

29-2232784 

9-4875182 

•001170960 

855 

731025 

625026375 

29-2403830 

9-4912200 

•001169591 

856 

732736 

627222016 

29-2574777 

9-4949188 

•001168224 

857 

734449 

629422793 

29-2745623 

9-4986147 

•001166861 

858 

736164 

631628712 

29-2916370 

9-5023078 

•001165501 

859 

737881 

633839779 

29-3087018 

9-5059980 

•001164144 

860 

739600 

636056000 

29-3257566 

9-5096854 

•001162791 

861 

741321 

638277381 

29-3428015 

9*5133699 

•001161440 

862 

743044 

640503928 

29-3598365 

9-5170515 

•001160093 

863 

744769 

642735647 

29-3768616 

9-5207303 

•001158749 

864 

746496 

644972544 

29-3938769 

9*5244063 

•001157407 

865 

748225 

647214625 

29*4108823 

9-5280794 

•001156069 

866 

749956 

649461896 

29-4278779 

9-5317497 

•001154734 

867 

751689 

651714363 

29-4448637 

9*5354172 

•001153403 

868 

753424 

653972032 

29-4618397 

9-5390818 

•001152074 

869 

755161 

656234909 

29-4788059 

9-5427437 

•001150748 

870 

756900 

658503000 

29-4957624 

9-5464027 

•001149425 

871 

758641 

660776311 

29-5127091 

9-5500589 

•001148106 

872 

760384 

663054848 

29-5296461 

9-5537123 

•001146789 

873 

762129 

665338617 

29-5465734 

9-5573630 

•001145475 

874 

763876 

667627624 

29-5634910 

9-5610108 

•001144165 

875 

765625 

669921875 

29-5803989 

9-5646559 

•001142857 

876 

767376 

672221376 

29-5972972 

9-5682782 

•001141553 

877 

769129 

674526133 

29-6141858 

9-5719377 

•001140251 

878 

770884 

676836152 

29-6310648 

9-5755745 

•001138952 

879 

772641 

679151439 

29-6479342 

9-5792085 

•001137656 

880 

774400 

681472000 

29-6647939 

9-5828397 

•001136364 

881. 

776161 

683797841 

29-6816442 

9-5864682 

•001135074 

882 

777924 

686128968 

29-6984848 

9-5900937 

•001133787 

883 

779689 

688465387 

29-7153159 

9-5937169 

•001132503 

884 

781456 

690807104 

29-7321375 

9-5973373 

•001131222 














99 


Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

V Roots. 

■$/ Roots. 

Reciprocals. 

885 

783225 

693154125 

29*7489496 

9-6009548 

•001129944 

886 

784996 

695506456 

29-7657521 

9-6045696 

•001128668 

887 

786769 

697864103 

29-7825452 

9-6081817 

•001127396 

888 

788544 

700227072 

29-7993289 

9-6117911 

•001126126 

889 

790321 

702595369 

29-8161030 

9-6153977 

•001124859 

89« 

792100 

704969000 

29-8328678 

9-6190017 

•001123596 

891 

793881 

707347971 

29-8496231 

9-6226030 

•001122334 

892 

795664 

707932288 

29-8663690 

9-6262016 

•001121076 

893 

797449 

712121957 

29-8831056 

9-6297975 

•001119821 

894 

7419236 

714516984 

29-8998328 

9-6333907 

•001118568 

895 

801025 

716917375 

29-9165506 

9-6369812 

•001117818 

896 

802816 

719323136 

29-9332591 

9-6405690 

•001116071 

897 

804609 

721734273 

29-9499583 

9-6441542 

•001114827 

898 

806404 

724150792 

29-9666481 

9-6477367 

•001113586 

899 

808201 

726572699 

29-9833287 

9-6513166 

•001112347 

900 

810000 

729000000 

30-0000000 

9-6548938 

•001111111 

901 

811801 

731432701 

30-0166621 

9-6584684 

•001109878 

902 

813604 

733870808 

30-0333148 

9-6620403 

•001108647 

903 

815409 

736314327 

30-0499584 

9-6656096 

•001107420 

904 

817216 

738763264 

30-0665928 

9-6691762 

•001106195 

905 

819025 

•741217625 

30-0832179 

9-6727403 

•001104972 

906 

820836 

743677416 

30-0998339 

9-6763017 

•001103753 

907 

822649 

746142643 

30-1164407 

9-6798604 

•001102536 

908 

824464 

748613312 

20-13303S3 

9-6834166 

•001101322 

909 

826281 

751089429 

30-1496269 

9-6S69701 

•001100110 

910 

828100 

753571000 

30-1662063 

9-6905211 

•001098901 

911 

829921 

75605S031 

30-1827765 

9-6940694 

•001097695 

912 

831744 

758550828 

30-1993377 

9-6976151 

•001096491/ 

913 

833569 

761048497 

30-2158899 

9-7011583 

•001095290 

914 

83.>>39 6 

763551944 

30-2324329 

9-7046989 

•001094092. 

915 

837225 

786060875 

30-2489669 

9-7082369 

•001092896 

916 

839056 

768575296 

30-2654919 

9-7117723 

•001091703 

917 

840889 

771095213 

30-2820079 

9-7153051 

•001090513 

918 

842724 

773620632 

30-2985148 

9-7188354 

•001089325 

919 

844561 

776151559 

30-3150128 

9-7223631 

•001088139 

920 

846400 

778688000 

30-3315018 

9-7258883 

•001086957 

921 

848241 

781229961 

30-3479818 

9.7294109 

•001085776 

922 

850084 

783777448 

30-3644529 

9-7329309 

•001084599 

923 

851929 

786330467 

30-3809151 

9-73644S4 

•001083423 

924 

853776 

788889024 

30-3973683 

9-7399634 

•001082251 

925 

855625 

791453125 

30-4138127 

9-7434758 

•001081081 

926 

857476 

794022776 

30-4302481 

9-7469S57 

•001079914 

927 

859329 

796597983 

30-4466747 

9-7504930 

•001078749 

928 

861184 

799178752 

30-4630924 

9-7539979 

•001077586 

929 

863041 

801765089 

30-4795013 

9-7575002 

•001076426 

930 

864900 

804357000 

30-4959014 

9-7610001 

•001075269 

931 

866761 

806954491 

30-5122926 

9-7644974 

•001074114 

932 

868624 

809557568 

30-5286750 

9-7679922 

•001072961 

933 

870489 

812166237 

30-5450487 

9-7714845 

•001071811 

934 

872356 

814780504 

30-5614136 

9-7749743 

•001070664 

935 

874225 

817400375 

30-5777697 

9-7784616 

•001069519 

936 

876096 

820025856 

30-5941171 

9-7819466 

•001068376 
























Table of Squares, Cubes, Square and Cube Rocts. 


Number. 

Squares. 

Cubes. 

r 

V Boots. 

V Boots. 

Reciprocals. 

937 

877969 

S22656953 

30-6104557 

9-7854288 

•001067236 

938 ■ 

879844 

825293672 

30-6267857 

9-7889087 

•001066098 

939 

881721 

827936019 

30-6431069 

9-7923861 

•001064963 

940 

883600 

830584000 

30-6594194 

9-7958611 

•001063830 

941 

885481 

833237621 

30-6757233 

9-7993336 

•001062699 

942 

887364 

S358968S8 

30-6920185 

9-8028036 

•001061571 

943 

889249 

838561807 

30-7083051 

9-8062711 

•001060445 

944 

891136 

841232384 

30-7245830 

9-8097362 

•001059322 

945 

893025 

843908625 

30-7408523 

9-8131989 

•001058201 

946 

894916 

846590536 

30-7571130 

9-8166591 

•#01057082 

947 

896809 

849278123 

30-7733651 

9-8201169 

•001055966 

948 

898704 

851971392 

30-7896086 

9-8235723 

•001054852 

949 

900601 

854670349 

30-8058436 

9-8270252 

•001053741 

950 

902500 

857375000 

30-8220700 

9-8304757 

•001052632 

951 

904401 

860085351 

30-8382879 

9-8339238 

•001051525 

952 

906304 

862801408 

30-8544972 

9-8373695 

•001050420 

953 

908209 

865523177 

30-8706981 

9-8408127 

•001049318 

954 

910116 

868250664 

30-8868904 

9-8442536 

•001048218 

955 

912025 

870983875 

30-9030743 

9-8476920 

•001047120 

956 

913936 

873722816 

30-9192477 

9-8511280 

•001046025 

957 

915849 

876467493 

30-9354166 

9-8545617 

•001044932 

958 

917764 

879217912 

30-9515751 

9-8579929 

•001043841 

959 

919681 

881974079 

30-9677251 

9-8614218 

•001042753 

960 

921600 

884736000 

30-9838668 

9-8648483 

•001041667 

961 

923521 

887503681 

31-0000000 

9-8682724 

•001040583 

962 

925444 

890277128 

31-0161248 

9-8716941 

•001039501 

963 

927369 

893056347 

31-0322413 

9-8751135 

•001038422 

964 

929296 

895841344 

31-0483494 

9-8785305 

•001037344 

965 

931225 

898632125 

31-0644491 

9-8819451 

•001036269. 

966 

933156 

901428696 

31-0805405 

9-8853574 

•001035197 

967 

935089 

904231063 

31-0966236 

9-8887673 

•001034126 

968 

937024 

907039232 

31-1126984 

9-8921749 

•001033058 

969 

938961 

909853209 

31-1287648 

9-8955801 

•001031992 

970 

940900 

912673000 

31-1448230 

9-8989830 

•001030928 

971 

942841 

915498611 

31-1608729 

9-9023835 

•001029866 

972 

944784 

918330048 

31-1769145 

9-9057817 

•001028807 

973 

946729 

921167317 

31-1929479 

9-9091776 

•001027749 

974 

948676 

924010424 

31-2089731 

9-9125712 

•001026694 

975 

950625 

926859375 

31-2249900 

9-9159624 

•001025641 

976 

952576 

929714176 

31-2409987 

9-9193513 

•001024590 

977 

954529 

932574833 

31-2569992 

9-9227379 

•001023541 

978 

956484 

935441352 

31-2729915 

9-9261222 

•001022495 

979 

958441 

938313739 

31-2889757 

9-9295042 

•001021450 

980 

960400 

941192000 

31-3049517 

9-9328839 

•001020408 

981 

962361 

944076141 

31-3209195 

9-9362613 

•001019168 

982 

964324 

946966168 

31-3368792 

9-9396363 

•001018330 

983 

966289 

949862087 

31-3528308 

9-9430092 

•001017294 

9S4 

968256 

952763904 

31-3687743 

9-9463797 

•001016260 

985 

970225 

955671625 

31-3847097 

9-9497479 

•001015228 

986 

972196 

958585256 

31-4006369 

9-9531138 

•001014199 

987 

974169 

961504803 

31-4165561 

9-9564775 

•001013171 

988 

976144 

964430272 : 31-4324673 

9-9598389 

•O01012146 






















Tabu, of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

989 

978121 

967361669 

990 

980100 

970299000 

991 

982081 

973242271 

992 

984064 

976191488 

993 

986049 

979146657 

994 

988036 

982107784 

995 

990025 

985074875 

996 

992016 

988047936 

997 

994009 

991026973 

998 

996004 

994011992 

999 

998001 

997002999 

1000 

1000000 

1000000000 

1001 

1002001 

1003003001 

1002 

1004004 

1006012008 

1003 

1006009 

1009027027 

1004 

1008016 

1012048064 

1005 

1010025 

1015075125 

1006 

1012036 

1018108216 

1007 

1014049 

1021147343 

1008 

1016064 

1024192512 

1009 

1018081 

1027243729 

1010 

1020100 

1030301000 

1011 

1022121 

1033364331 

1012 

1024144 

1036433728 

1013 

1026169 

1039509197 

1014 

1028196 

1042590744 

1015 

1030225 

1045678375 

1016 

1032256 

1048772096 

1017 

1034289 

1051871913 

1018 

1036324 

1054977832 

1019 

1038361 

1058089859 

1020 

1040400 

1061208000 

1021 

1042441 

1064332261 

1022 

1044484 

1067462648 

1023 

1046529 

1070599167 

1024 

1048576 

1073741824 

1025 

1050625 

1076890625 

1026 

1052676 

1080045576 

1027 

1054729 

1083206683 

1028 

1056784 

1086373952 

1029 

1058841 

1089547389 

1030 

1060900 

1092727000 

1031 

1062961 

1095912791 

1032 

1065024 

1099104768 

1033 

1067089 

1102302937 

1034 

1069156 

1105507304 

1035 

1071225 

1108717875 

1036 

1073296 

1111934656 

1037 

1075369 

1115157653 

1038 

1077444 

1118386872 

1039 

1079521 

1121622319 

1040 

1081600 

1124864000 


V i loots. 

31-4483704 

31-4642654 

31-4801525 

31-4960315 

31-5119025 

31-5277655 

31-5436206 

31-5594677 

31-5753068 

31-5911380 

31-6069613 

31-6227766 

31-6385840 

31-6543866 

31-6701752 

31-6859590 

31-7017349 

31-7175030 

31-7332633 

31-7490157 

31-7647603 

31-7804972 

31-7962262 

31-8119474 

31-8276609 

31-8433666 

31-8590646 

31-8747549 

31-8904374 

31-9061123 

31-9217794 

31-9374388 

31-9530906 

31-9687347 

31- 9843712 

32- 0000000 
32-0156212 
32-0312348 
32-0468407 
32-0624391 
32-0780298 
32-0936131 
32-1091887 
32-1247568 j 
32-1403173 
32-1558704 
32-1714159 
32-1869539 
32-2024844 j 
32-2180074 j 
32-2335229 ' 
32-2-190310 ! 


\/ Roots. 

9-9631981 
9-9665549 
9-9699055 
9-9732619 
9-9766120 
9-9799599 
9-9833055 
9-9866488 
9-9899900 
9-9933289 
9-9966656 
10-0000000 
10-0033222 
10-0066622 
10-0099899 
10 0133155 
10-0166389 
10-0199601 
10-0232791 
10-0265958 
10-0299104 
10-0332228 
10-0365330 
10-0398410 
10-0431469 
10-0464506 
10-0497521 
10-0530514 
10-0563485 
10-0596435 
10-0629364 
10-0662271 
10-0695156 
10-0728020 
10-0760863 
10-0793684 
10-0826484 
10-0859262 
10-0892019 
10-0924755 
10-0957469 
10-0990163 
10-1022835 
10-1055487 
10*10SS117 
10-1120726 
10-1153314 
10-1185882 
10-1218428 
10-1250953 
10-1283457 
10-1315941 


I'M 


Reciprocals. 

*001011122 

*001010101 

•001009082 

•001008065 

•001007049 

•001006036 

•001005025 

•001004016 

•001003009 

•001002004 

•001001001 

■001000000 

•0009990010 

•0009980040 

•0009970090 

•0009960159 

•0009950249 

•0009940358 

•0009930487 

•0009920635 

•0009910803 

•0009900990 

•0009891197 

•0009881423 

•0009871668 

•0009861933 

•0009852217 

•0009842520 

•0009S32842 

•0009823183 

•0009813543 

•0009803922 

•0009794319 

•0009784736 

•0009775171 

•0009765625 

•0009756098 

•0009746589 

•0009737098 

•0009727626 

•0009718173 

•0009708738 

•0009699321 

•0009689922 

•0009680542 

•0009671180 

•0009661836 

•0009652510 

•0009643202 

•0009633911 

•0U09624639 

•0009615385 




















102 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

■ 

Squares. 

Cubes. 

sj Roots. 

Roots. 

Reciprocals. 

1041 

1083681 

1128111921 

32*2645316 

10-1348403 

•0009606148 

1042 

1085764 

1131366088 

32-2800248 

10-1380S45 

•0009596929 

1043 

10S7849 

1134626507 

32-2955105 

10-1413266 

•0009587728 

1044 

1089936 

1137893184 

32-3109888 

10-1445667 

•0009578544 

1045 

1092025 

1141166125 

32-3264598 

10-1478047 

•0009569378 

1046 

1094116 

1144445336 

32-3419233 

10-1510406 

•0009560229 

1047 

1096209 

1147730823 

32-3573794 

10-1542744 

•0009551098 

1048 

1098304 

1151022592 

32-3728281 

10-1575062 

•0009541985 

1049 

1100401 

1154320649 

32-3882695 

10-1607359 

•0009532888 

1050 

1102500 

1157625000 

32-4037035 

10-1639636 

•0009523810 

1051 

1104601 1160935651 

32-4191301 

10-1671893 

•0009514748 

1052 

1106704 

1164252608 

32-4345495 

' 10-1704129 

•0009505703 

1053 

1108809 

1167575877 

32-4499615 

10-1736344 

•0009496676 

1054 

1110916 

1170905464 

32-4653662 

10-1768539 

•0009487666 

1055 

1113025 

1174241375 

32-4807635 

10-1800714 

•0009478673 

1056 

1115136 

1177583616 

32-4961536 

10-1832868 

•0009469697 

1057 

1117249 

1180932193 

32-5115364 

10-1865002 

•0009460738 

1058 

1119364 

1184287112 

32-5269119 

10-1897116 

•0009451796 

1059 

1121481 

1187648379 

32-5422802 

10-1929209 

•0009442871 

1060 

1123600 

1191016000 

32-5576412 

10-1961283 

•0009433962 

1061 

1125721 

1194389981 

32-5729949 

10-1993336 

•0009425071 

1062 

1127844 

1197770328 

32-5883415 

10-2025369 

*0009416196 

1063. 

1129969 

1201157047 

32-6035807 

10-2057382 

•0009407338 

1064 

1132096 

1204550144 

32-6190129 

10-2089375 

•0009398496 

1065 

1134225 

1207949625 

32-6343377 

10-2121347 

•0009389671 

1066 

1136356 

1211355496 

32-6496554 

10-2153300 

•0009380863 

1067 

1138489 

1214767763 

32-6649659 

10-2185233 

•0009372071 

1068 

1140624 

1218186432 

32-6802693 

10-2217146 

•0009363296 

1069 

1142761 

1221611509 

32-6955654 

10-2249039 

•0009354537 

1070 

1144900 

1225043000 

32-7108544 

10-2280912 

•0009345794 

1071 

1147041 

1228480911 

32-7261363 

10-2312766 

*0009337068 

1072 * 

1149184 

1231925248 

32-7414111 

10-2344599 

•0009328358 

1073 

1151329 

1235376017 

32-7566787 

10-2376413 

•0009319664 

1074 

1153476 

1238833224 

32-7719392 

10-2408207 

•0009310987 

1075 

1155625 

1242296875 

32-7871926 

10-2439981 

•0009302326 

1076 

1157776 

1245766976 

32-8024398 

10-2471735 

•0009293680 

1077 

1159929 

1249243533 

32-8176782 

10-2503470 

•0009285051 

1078 

1162084 

1252726552 

32-8329103 

10-2535186 

•0009276438 

1079 

1164241 

1256216039 

32-8481354 

10-2566881 

•0009267841 

1080 

1166400 

1259712000 

32-8633535 

10-2598557 

*0009259259 

1081 

1168561 

1263214441 

32-8785644 

10-2630213 

•0009250694 

1082 

1170724 

1266723368 

32-8937684 

10-2661850 

*0009242144 

1083 

1172889 

1270238787 

32-9089653 

10-2693467 

•0009233610 

1084 

1175056 

1273760704 

32-9241553 

10-2725065 

•0009225092 

1085 

1177225 

1277289125 

32-9393382 

10-2756644 

•0009216590 

1086 

1179396 

1280824056 

32-9545141 

10-2788203 

*0009208103 

1087 

1181569 

1284365503 

32-9696830 

10-2819743 

•0009199632 

1088 

1183744 

1287913472 

32-9848450 

10-2851264 

•0009191176 

1089 

1185921 

1291467969 

33-0000000 

10-2882765 

•0009182736 

1090 

1188100 

1295029000 

33-0151480 

10-2914247 

•0009174312 

1091 , 

1190281 

1298596571 

33-0302891 

10-2945709 

•0009165903 

1092 

1192464 

1302170688 1 

33-0454233 1 

10-2977153 ! 

•0009157509 

j 



























Table op Squares, Cubes, Square and Cube Roots. 


103 


Number. 

Squares. 

Cubes. 

V^Roots. 

Roots. 

Reciprocals. 

1093 

1194649 

1305751357 

33-0605505 

10-3008577 

•0009149131 

1094 

1196836 

1309338584 

33-0756708 

10-3039982 

•0009140768 

1095 

1199025 

1312932375 

33-0907842 

10-3071368 

•0009132420 

1096 

1201216 

1316532736 

33-1058907 

10-3102735 

•0009124008 

109? 

1203409 

1320139673 

33-1209903 

10-3134083 

•0009115770 

1098 

1205604 

1323753192 

33-1360830 

10-3165411 

•0009107468 

1099 

1207801 

1327373299 

33-1511689 

10-3196721 

•0009099181 

1100 

1210000 

1331000000 

33-1662479 

10-3228012 

•0009090909 

1101 

1212201 

1334633301 

33-1813200 

10-3259284 

•0009082652 

1102 

1214404 

1338273208 

33-1963853 

10-3290537 

•0009074410 

1103 

1216609 

1341919727 

33*2114438 

10-3321770 

•0009066183 

1104 

1218816 

1345572864 

33-2266955 

10-3352985 

•0009057971 

1105 

1221025 

1349232625 

33-2415403 

10-3384181 

•0009049774 

1106 

1223236 

1352899016 

33-2565783 

10-3415358 

•0009041591 

1107 

1225449 

1356572043 

33-2716095 

10-3446517 

•0009033424 

1108 

1227664 

1360251712 

33*2866339 

10-3477657 

•0009025271 

1109 

1229881 

1363938029 

33-3016516 

10-3508778 

•0009017133 

1110 

1232100 

1367631000 

33-3166625 

10-3539880 

•0009009009 

1111 

1234321 

1371330631 

33*3316666 

10-3570964 

•0009000900 

1112 

1236544 

1375036928 

33-3466640 

10-3602029 

•0008992806 

1113 

1238769 

1378749897 

33-3616546 

10-3633076 

•0008984726 

1114 

1240996 

1382469544 

33-3766385 

10-3664103 

•0008976661 

1115 

1243225 

1386195875 

33-3916157 

10-3695113 

•0008968610 

1116 

1245456 

1389928896 

33-4065862 

10-3726103 

•0008960753 

1117 

1247689 

1393668613 

33-4215499 

10-3757076 

•0008952551 

1118 

1249924 

1397415032 

33*4365070 

10-3788030 

•0008944544 

1119 

1252161 

1401168159 

• 33-4514573 

10-3818965 

•0008936550 

1120 

1254400 

1404928000 

33-4664011 

10-3849882 

•0008928571 

1121 

1256641 

1403694561 

33-4813381 

10-3880781 

•0008960607 

1122 

1258884 

1412467848 

33-4962684 

10-3911661 

•0008922656 

1123 

1261129 

1416247867 

33-5111921 

10-3942527 

•0008904720 

1124 

1263376 

1420034624 

33*5261092 

10-3973366 

•0008896797 

1125 

1265625 

1423828125 

33-5410196 

10-4004192 

•0008888889 

1126 

1267876 

1427628376 

33-5559234 

10-4034999 

•0008880995 

1127 

1270129 

1431435383 

33-5708206 

10-4065787 

•0008873114 

1128 

1272384 

1435249152 

33-5857112 

10-4096557 

•0008865248 

1129 

1274641 

1439069689 

33-6005952 

10-4127310 

•0008857396 

1130 

1276900 

1442897000 

33-6154726 

10-4158044 

•0008849558 

1131 

1279161 

1446731091 

33-6303434 

10-4188760 

•0008841733 

1132 

1281424 

1450571968 

33-6452077 

10-4219458 

•0008833922 

1133 

1283689 

1454419637 

33-6600653 

10-4250138 

•0008826125 

1134 

1285956 

1458274104 

33-6749165 

10-4280800 

•0008818342 

1135 

1288225 

1462135375 

33-6897610 

10-4311443 

•0008810573 

1136 

1290496 

1466003456 

83-7045991 

10-4342069 

•0008802817 

1137 

1292769 

1469878353 

33-7174306 

10-4372677 

•0008795075 

1138 

1295044 

1473760072 

33-7340556 

10-4403677 

•0008787346 

1139 

1297321 

1477648619 

33-7490741 

10-4433839 

•0008779631 

1140 

1299600 ! 

1481544000 

33-7638860 

10-4464393 

•0008771930 

1141 

1301881; 

1485446221 

33-7786915 

10-4494929 

•0008764242 

1142 

1304164 ! 

1489355288 

33-7934905 

10-4525448 

•0008756567 

1143 

1306449! 

1493271207 

33-8082830 

10-4555948 

•0008748906 

1144 

1308736 

1497193984 

33-8230691 1 

10-4586431 

•0008741259 















104 


Table of Squares, Cubes, Square and Cube Roots. 


Number 

Squares. 

Cutes. 

V Roots. 

S? Roots. 

Reciprocals. 

2145 

1311025 

1501123625 

33-8378486 

10-4616896 

•0008733624 

1140 

1313316 

1505060136 

33*8526218 

10-4647343 

•0008726003 

1147 

1315609 

1509003523 

33-8673884 

10-4677773 

*0008718396 

1148 

1317904 

1512953792 

33*8821487 

10-4708158 

•0008710801 

1149 

1320201 

1516910949 

33-8969025 

10-4738579 

-0008703220 

1150 

1322500 

1520875000 

33*9116499 

10-4768955 

•0008695652 

1151 

1324801 

1524845951 

33-9263909 

10-4799314 

•0008688097 

1152 

1327104 

1528823808 

33-9411255 

10-4829656 

*0008680556 

1153 

132y409 

1532808577 

33-9558537 

10-4859980 

-0008673027 

1154 

1331716 

1536800264 

33-9705755 

10-4890286 

•0008665511 

1155 

1334025 

1540798875 

33-9852910 

10-4920575 

•0008658009 

1156 

1336336 

1544804416 

34-0000000 

10-4950847 

•0008650519 

1157 

1338649 

1548816893 

34-0147027 

10-4981101 

•0008643042 

1158 

134096-1 

1552836312 

34-0293990 

10-5011337 

•0008635579 

1159 

1343281 

1556862679 

34-0440890 

10-5041556 

•0008628128 

1160 

1345600 

1560896000 

34-0587727 

10-5071757 

•0008620690 

1161 

1347921 

1564936281 

34-0734501 

10*5]01942 

•0008613264 

1162 

1350244 

1568983528 

34-0881211 

10-5132109 

•0008605852 

1163 

1352569 

1573037747 

34*0127858 

10-5162259 

•0008598452 

1164 

1354896 

1577098944 

34-1174442 

10-5192391 

•0008591065 

1165 

1357225 

1581167125 

34-1320963 

10-5222506 

•0008583691 

1166 

1359556 

1585242296 

34-1467422 

10-5252604 

•0008576329 

1167 

1361889 

1589324463 

34-1613817 

10-5282685 

•0008568980 

1168 

1364224 

1593413632 

34-1760150 

10-5312749 

•0008561644 

1169 

1366561 

1597509809 

34-1906420 

10-5342795 

•0008554320 

1170 

1368900 

1601613000 

34*2052627 

10-5372825 

•0008547009 

1171 

1371241 

1605723211 

34-2198773 

10-5402837 

•0008539710 

1172 

1373584 

1609840448 

34-2344S55 

10-5432832 

•0008532423 

1173 

1375929 

1613964717 

34*2490875 

10-^462810 

•0008525149 

1174 

1378276 

1618096024 

34*2636834 

10-5492771 

•0008517888 

1175 

1380625 

1622234375 

34-2782730 

10-5522715 

•0008510638 

1176- 

1382976 

1626379776 

34-2928564 

10-5552642 

•0008503401 

1177 

1385329 

1630532233 

34*3074336 

10-5582552 

•0008496177 

1178 

1387684 

1634691752 

34-3220046 j 

10-5612445 

•0008488964 

1179 

1390041 

1638858339 

34-3365694 

10-5642322 

•0008481764 

1180 

1392400 

1643032000 

34-3511281 

10-5672181 

•0008471576 

1181 

1394761 

1647212741 

34-3656805 

10-5702024 

•0008467401 

1182 

1397124 

1651400568 

34-3802268 

10-5731849 

•0008460237 

1183 

1399489 

1655595487 

34*3947670 1 

10-5761658 

•0008453085 

1184 

1401856 

1659797504 

34-4093011 

10-5791449 

•0008445946 

1185 

1404225 

1664006625 

34-4238289 

10-5821225 

•0008438819 

11S6 

1406596 

1668222856 

34-4383507 

10-5850983 

•0008431703 

1187 

1408969 

1672446203 

34-452S663 

10-5880725 

*0008421600 

1188 

1411344 

1676676672 

34-4673759 

10-5910450 

•0008417508 

1189 

1413721 

1680914629 

34-4818793 

10-5940158 1 

•0008410429 

1190 

1416100 

1685159000 

34-4963766 

10-5969850 

•0008403361 

1191 

1418481 

1689410871 

34-5108678 

10-5999525 

•0008396308 

1192 

1420864 

1693669S88 

34-5253530 

10-6029184 ! 

•0008389262 

1193 i 

1423249 

1697936057 

34-5398321 j 

10-6058826 

•0008382320 

1194 

1425636 

1702209384 

34-5543051 

10-6088451 

•0008375209 

1195 

1428025 

17064S9875 

34-5687720 

10-611S060 

•0008368201 

1196 1 

1430416 

1710777536' 

34-5832329 1 

10-6147652 

•0008361204 






























Table of Squares, Cubes,- Square and Cube Hoots. jqj 


- 

Numter. 

Squares. 

Cube*. 

| y/~Roota. 

y/ Roots. 

Reciprocals. 

1197 

1432809 

,1735072373 

34-5976879 

10-6177228 

•0008354219 

1198 

1435204 

'1719374392 

j 34-6121366 

10-6206788 

•0008347245 

1199 

1437601 

1723683599 

34*6265794 

10-6236331 

•0008340284 

1200 

1440000 

1728000000 

34-6410162 

10-6265857 

*0008333333 

1201 

1442401 

1732323601 

34-6554469 

10-6295367 

*0008326395 

3202 

1444804 

1736654408 

34*6698716 

10-6324860 

•0008319468 

1203 

1447209 

1740992427 

34*6842904 

10-6354338 

*0008312552 

1204 

1449616 

1745337664 

34-6987031 

10-6383799 

*0008305648 

1205 

1452025 

11749690125 

34-7131099 

10-6413244 

•0008298755 

1206 

1454436 

'1754049816 

34-7275107 

10-6442672 

•0008291874 

1207 

1456849 

11758416743 

34-7419055 

10-6472085 

•0008285004 

1208 

1459264 

1762790912 

34*7562944 

10-6501480 

•0008278146 

1209 

1461681 

1767172329 

34-7706773 

10-6530860 

•0008271299 

1210 

1464300 

1771561000 

34-7850543 

10-6560223 

•0008264463 

1211 

1466521 

1775956931 

34*7994253 

10-6589570 

•0008257638 

1212 

1468944 

1780360128 

34-8137904 

10-6618902 

•0008250S25 

1213 

1471369 

1784770597 

34*8281495 

10-6648217 

•0008244023 

1214 

1473796 

1789188344 

34-8425028 

10-6677516 

*0008237232 

1215 

1476225 

1793613375 

34-8568501 

10-6706799 

*0008230453 

1216 

1478656 

1798045696 

34-87H915 

10-6736066 

*0008223684 

1217 

1481089 

1802485313 

34-8855271 

10-6765317 

•0008216927 

1218 

1483524 

1806932232 

34-8998567 

10-6794552 

•0008210181 

1219 

1485961 

1811386459 

34-9141805 

10-6823771 

•0008203445 

1220 

1488400 

1815848000 

34*9284984 

10-6852973 

•0008196721 

1221 

1490841 

1820316861 

34-9428104 

10-6882160 

•0008190008 

1222 

1493284 

j1824793048 

34-9571166 

10-6911331 

•0008183306 

1223 

1495729 

1829276567 

34*9714169 

10-6940486 

•0008176615 

1224 

1498176 

1833764247 

34-9857114 

10-6969625 

•0008169935 

1225 

1500625 

[1838265625 

35-0000000 

10-6998748 

*0008163265 

1226 

1503276 

1842771176 

35'0142828 

10-7027855 

•0008156607 

1227 

1505529 

1847284083 

35-0285598 

10-7056947 

•0008149959 

1228 

3507984 

1851804352 

35-0428309 

10-7086023 

•0008143322 

1229 

1510441 

1856331989 

35-0570963 

10-7115083 

•0008136696 

. 1230 

1512900 

1860867000 

35*0713558 

10-7144127 

•0008130081 

1231 

3515361 

1865409391 

35-0856096 

10-7173155 

•0008123477 

1232 

1517824 

1869959168 

35-0998575 

10-7202168 

*0008116883 

1233 

1520289 

1874516337 

35-1140997 

10-7231165 

•0008110300 

1234 

1522756 

18790S0904 

351283361 

10-7260146 

•0008103728 

1235 

1525225 

1883652875 

35*1425668 

10-7289112 

•0008097166 

1236 

1527696 

1888232256 

35-1567917 

10-7318062 

•0008090615 

1237 

1530169 

1892819053 

35*1710108 

10-7346997 

•0008084074 

1238 

1532644 

1897413272 

35*1852242 

10-7375916 

•000S077544 

1239 

1535121 

1902014919 

35*1994318 

10-7404819 

•0008071025 

1240 

1537600 

1906624000 

35-2136337 

10-7433707 

•0008064516 

1241 

1540081 

1911240521 

35*2278299 

10-7462579 

•0008058018 

1242 

1542564 

1915864488 

35*2420204 

10-7491436 

•0008051530 

1243 

1545049 

1920495907 

35-2562051 

10-7520277 

•0008045052 

1244 

1547536 

1925134784 

35*2703842 

10-7549103 

•0008038585 

1245 

1550025 

1929781125 

35-2845575 

10-7577913 

•0008032129 

1246 

1552536 

1934434936 

35-2987252 

10-7606708 

•0008025682 

1247 

1555009 

1939096223 

35-3128872 

10-' 7 6354S8 

•0008019246 

1248 

1557504' 

1943764992 

35-3270435 

10-7664252 

•0008012821 























106 


Table of Squares, Cubes, Square and Cube Hoots 


Number 

Squares. 

Cubes. 

V Roots. 

■ <y Roots. 

Reciprocals. 

1249 

1560001 

1948441249 

35*3411941 

10-7693001 

•0008006405 

1250 

1562500 

1953125000 

35-3553391 

10-7721735 

•0008000000 

1251 

1565001 

1957816251 

35-3694784 

10-7750453 

•0007993605 

1252 

1567504 

1962515008 

35-3836120 

10-7779156 

•0007987220 

1253 

1570009 

1967221277 

35-3977400 

10-7807843 

•0007980846 

1254 

1572516 

1971935064 

35-4118624 

10-7836516 

•000797448.2 

1255 

1575025 

1976656375 

35-4259792 

10-7865173 

•0007968127 

1256 

1577536 

1981385216 

35-4400903 

10-7893815 

•0007961783 

1257 

1580049 

1986121593 

35-4541958 

10-7922441 

•0007955449 

125S 

1582564 

1990865512 

35-4682957 

10-7951053 

•0007949126 

1259 

1585081 

1995616979 

35-4823900 

10-7979649 

0007942812 

1260 

1587600 

2000376000 

35-4964787 

10-8008230 

•0007936508 

1261 

1590121 

2005142581 

35-5105618 

10-8036797 

•0007930214 

1262 

1592644 

2009916728 

35-5246393 

10-8065348 

•0007923930 

1263 

1595169 

2014698447 

35-5387113 

10-8093884 

•0007917656 

1264 

1597696 

2019487744 

35-5527777 

10-8122404 

•0007911392 

1265 

1600225 

2024284625 

35*5668385 

10-8150909 

•0007905138 

1266 

1602756 

2029089096 

35-5808937 

10-8179400 

•0007898894 

1267 

1605289 

2033901163 

35-5949434 

10-8207876 

•0007892660 

1268 

1607824 

2038720832 

35-6089876 

10-8236336 

•0007886435 

1269 

1610361 

2043548109 

35-6230262 

10-8264782 

•0007880221 

1270 

1612900 

2048383000 

35-6370593 

10-8293213 

•0007874016 

1271 

1615441 

2053225511 

35-6510869 

10-8321629 

•0007867821 

1272 

1617984 

2058075648 

35-6651090 

10-8350030 

•0007861635 

1273 

1820529 

2062933417 

35-6791255 

10-8378416 

•0007855460 

1274 

1623076 

2067798824 

35-6931366 

10-8406788 

•0007849294 

1275 

1625625 

2072671875 

35-7071421 

10-8435144 

•0007843137 

1276 

1628176 

2077552576 

35-7211422 

10-8463485 

•0007836991 

1277 

1630729 

j 2082440933 

35-7351367 

10-8491812 

•0007830854 

1278 

1633284 

2087336952 

35-7491258 

10-8520125 

•0007824726 

1279 

1635841 

2092240639 

35-7631095 

10-8548422 

•0007818608 

12S0 

1638400 

2097152000 

35-7770876 

10-8576704 

•0007812500 

1281 

1640961 

2102071841 

35-7910603 

10-8604972 

•0007806401 

1282 

1643524 

2106997768 

35-8050276 

10-8633225 

•0007800312 

1283 

1646089 

2111932187 

35*8189894 

10-8661454 

•0007794232 

1284 

1648656 

2116874304 

35*8329457 

10-8689687 

•0007788162 

1285 

1651225 

2121824125 

35-8468966 

10-8717897 

•0007782101 

1286 

1653796 

2126781656 

35*8608421 

10-8746091 

•0007776050 

1287 

1656369 

2131746903 

35-8747822 

10-8774271 

•0007770008 

1288 

1658944 

2136719872 

35-8887169 

10-8802436 

•0007763975 

1289 

1661521 

2141700569 

35-9026461 

10-8830587 

•000*7757952 

1290 

1664100 

2146689000 

35-9165699 

10-8858723 

•0007751938 

1291 

1666681 

2151685171 

35-9304884 

10-8886845 

•0007745933 

1292 

1669264 

2156689088 

35-9444015 

10-8914952 

•0007739938 

1293 

1671849 

2161700757 

35-9583092 

10-S943044 

•0007733952 

1294 

1674436 

2166720184 

35-9722115 

10-8971123 

•0007727975 

1295 

1677025 

i2171747375 

35-9861084 

10-8999186 

•0007722008 

1296 

1679616 

2176782336 

36-0000000 

10-9027235 

•0007716049 

1297 

1682209 

2181825073 

36-0138862 

10-9055269 

•0007710100 

1298 

1684804 

2186875592 

36-0277671 

10-9083290 

•0007704160 

1299 

1687401 

2191933899 

36-0416426 

10-9111296 

•0007698229 

1300 

1690000 

2197000000 

36-0555128 

10-9139287 

•0007692308 






















107 


Table of Squares, Cubes, Square and Cube Root:: 


N umber. 

Squares. 

Cubes. 

Roots. 

4/ Roots. 

1301 

1692601 

2202073301 

36-0693776 

10-9167265 

1302 

1695204 

2207155608 

36-083237 I 

10-9195228 

1303 

1697809 

2212245127 

36-0970913 

10-9223177 

1304 

1700416 

2217342464 

36-11094 02 

10-9251111 

1305 

1703025 

2222447625 

36-1247837 

10-9279031 

1306 

1705636 

2227560616 

36-1386220 

10-9306937 

1307 

1708249 

2232681443 

36-1524550 

10-9334829 

1308 

1710864 

2237810112 

36-1662826 

10-9362706 

1309 

1713481 

2242946629 

36-1801050 

10*9390569 

1310 

1716100 

2248091000 

36-1939221 

10-9418418 

1311 

1718721 

2253243231 

36-2077340 

10-9446253 

1312 

1721344 

2258403328 

36-2215406 

10-9475074 

1313 

1723969 

2263571297 

36-2353419 

10-9501880 

1314 

1726596 

2268747144 

36-2491379 

10-9529673 

1315 

1729225 

2273930875 

36-2626287 

10-9557451 

1316 

1731856 

2279122496 

36-2767143 

10-9585215 

1317 

1734489 

2284322013 

36-2904246 

10-9612965 

1318 

1737124 

2289529432 

36-3042697 

10-9610701 

1319 

1739761 

2294744759 

36*3180396 

10-9668423 

1320 

1742400 

2299968000 

36-3318042 

10-9696131 

1321 

1745041 

2305199161 

36-3455637 

10-9723825 

1322 

1747684 

2310438248 

36*3*593179 

10-9751505 

1323 

1750329 

2315685267 

36-3730670 

10-9779171 

1324 

1752976 

2320940224 

36-3868108 

10-9806823 

1325 

1755625 

2326203125 

36-4005494 

10*9834462 

1326 

1758276 

2331473976 

36*4142829 

10-98620S6 

1327 

1760929 

2336752783 

36-4280112 

10-9889696 

1328 

1763584 

2342039552 

36-4417343 

10-9917293 

1329 

1766241 

2347334289 

36*4554523 

10-9944876 

1330 

1768900 

2352637000 

38 4691650 

10-9972445 

1331 

1771561 

2357947691 

36-4828727 

11-0000000 

1332 

1774224 

2363266368 

36-4965752 

11-0027541 

1333 

1776889 

2368593037 

36-5102725 

11-0055069 

1334 

1779556 

2373927704 

36-5239647 

11-0082583 

1335 

1782225 

2379270375 

36-5376518 

11-0110082 

1336 

1784896 

2384621056 

36-5513388 

11-0137569 

1337 

1787569 

2389979753 

36-5650106 

11-0165041 

1338 

1790244 

2395346472 

36-5786823 

11-0192500 

1339 

1792921 

2400721219 

36-5923489 

11-0219945 

1340 

1795600 

2406104000 

36-6060104 

11-0247377 

1341 

1798281 

2411494821 

36-6196668 

11-0274795 

1342 

1800964 

2416893688 

36-6333181 

11-0302199 

1343 

1803649 

2422300607 

36-6469144 

11-0329590 

1344 

1806336 

2^27715584 

36-6606056 

11-0356967 

1345 

1809025 

2133138625 

36-6742416 

11-0384330 

1346 

1811716 

2438569736 

36-6878726 

11-0411680 

1347 

1814409 

2444008923 

36-7014986 

11-0439017 

1348 

1817104 

2449456192 

36-7151195 

11-04 66339 

1349 

1819801 

2454911549 

36-7287353 

11-0493649 

1350 

1822500 

2460375000 

36-7423461 

11-0520945 

1351 

1825201 

2465846551 

36-7559519 

11-0548227 

1352 

1827904 2471326208 

36-7695526 

11-0575497 


Reciprocals. 

•0007686395 
0007680492 
0007674579 
0007668712 
0007662835 
0007656968 
0007651109 
0007645260 
0007639419 
0007633588 
0007627765 
0007621951 
0007616446 
0007610350 
0007604563 
0007598784 
0007593014 
0007587253 
0007581501 
0007575758 
0007570023 
0007564297 
0007558579 
0007552870 
0007547170 
0007541478 
0007535795 
0007530120 
0007524454 
0007518797 
0007513148 
0007507508 
0007501875 
0007496252 
0007490637 
0007485030 
0007479432 
0007473842 
0007468260 
0007462687 
00074 57*122 
0007451565 
0007446016 
0007440476 
0007434944 
0007429421 
0007423905 
0007418398 
0007412898 
0007407407 
0007401924 
0007396450 


l 























103 Table op Squares, Cubes, Square and Cube Roots 


Number 

Squares. 

| Cubes. 

V' Hoots. 

| $/ Roots. 

Reciprocals. 

1353 

1830609 

2476813977 

36-7831483 

! 11-0602752 

•0007390983 

1354 

1833316 

2482309864 

36-7967390 

11-0629994 

•0007385524 

1355 

1836025 

2487813875 

36-8103246 

11-0657222 

•0007380074 

1356 

1838736 

2493326016 

36-8239053 

11-0684437 

•0007374631 

1357 

1841449 

2498846293 

36‘8374809 

11-0711639 

•0007369197 

1358 

1S44164 

2504374712 

36-8510515 

11-0738828 

•0007363770 

1359 

1846881 

2509911279 

36*8646172 

11-0766003 

•0007358352 

1360 

1849600 

2515456000 

36-8781778 

11-0793165 

•0007352941 

1361 

1852321 

2521008881 

36-8917335 

11-0820314 

•0007347539 

1362 

1855044 

2526569928 

36-9052842 

11-0847449 

•0007342144 

1363 

1857769 

2532139147 

36-9188299 

11-0874571 

•0007336757 1 

1364 

1860496 

2537716544 

36-9323706 

11-0901679 

•0007331378 

1365 

1863225 

2543302125 

36-9459064 

11-0928775 

•0007326007 

1366 

1865956 

2548895896 

36-9594372 

11-0955857 

•0007320644 

1367 

186S689 

2554497863 

36-9729631 

11-0982926 

•0007315289 

1368 

1871424 

256010S032 

36-9864840 

11-1009982 

•0007309942 

1369 

1874161 

2565726409 

37-0000000 

11-1037025 

•0007304602 

1370 

1876900 

2571353000 

37-0135110 

11-1064054 

•0007299270 

1371 

1879641 

2576987811 

37-0270172 

11-1091070 

•0007293946 

1372 

1882384 

2582630848 

37-0405184 

11-1118073 

•0007288630 

1373 

1885129 

2588282117 

37-0540146 

11-1145064 

•0007283321 

1374 

1887876 

2593941624 

37-0675060 

11-1172041 

•0007278020 

1375 

1890625 2599609375 

37-0899924 

11-1199004 

•0007272727 

1376 

1893376 

2605285376 

37-0944740 

11-1225955 

•0007267442 

1377 

1896129 

2610969633 

37-1079506 

11-1252893 

•0007262164 

1378 

1898884 

2616662152 

37-1214224 

11-1279817 

•0007256894 

1379 

1901641 

2622362939 

37-1348893 

11-1306729 

•0007251632 

1380 

1904400 

2628072000 

37-1483512 

11-1333628 

•0007246377 

1381 

1907161 

2633789341 

37-1618084 

11-1360514 

•0007241130 

1382 

1909924 

2639514968 

37-1752606 

11-1387386 

•0007235890 

1383 

1912689 

2645248887 

37-1887079 

11-1414246 

•0007230658 

1384 

1915456 

2650991104 

37-2021505 

11-1441093 

•0007225434 

1385 

1918225 

2656741625 

37-2155881 

11-1467926 

•0007220217 

1386 

1920996 

2662500456 

37-2290209 

11-1494747 

•0007215007 

1387 

1923769 

2668267603 

37-2424489 

11-1521555 

•0007209805 

1388 

1926544 

2674043072 

37-2558720 

11-1548350 

•0007204611 

1389 

1929321 

2679826869 

37-2692903 

11-1575133 

•0007199424 

1390 

1932100 

2685619000 

37-2827037 

11-1601903 

•0007194245 

1391 

1934881 

2691419471 

37-2961124 

11-1628659 

•00071S9073 

1392 

1937664 

2697228288 

37-3095162 

11-1655403 

•0007183908 

1393 

1940449 

2703045457 

37-3229152 

11-1682134 

•0007178751 

1394 

1943236 

2708870984 

37-3363094 

11-1708852 

•0007173601 : 

1395 

1946025 

2714704875 

37-3496988 

11-1735558 

•0007168459 

1396 

1948816 

2720547136 

37-3630834 

11-1762250 

•0007163324 

1397 

1951609 

2726397773 

37-3764632 

11-1788930 

•0007158196 

1398 

1954404 

2732256792 

37-3898382 

11-1815598 

•0007153076 

1399 

1957201 

2738124199 

37-4032084 

11-1842252 

•0007147963 

1400 

1960000 

2744000000 

37-4165738 

11-1868894 

•0007142857 

1401 

1962801 

2749884201 

37-4299345 

11-1895523 

•0007137759 

1402 

1965604 

2755776808 

37-4432904 

11-1922139 

•0007132668 

1403 

1968409 

2761677827 

37-4566416 

11-1948743 

•0007127584 

1404 

1971216 

2767587264 

37-4699880 

11-1975334 

•0007122507 






























Table of Squares, Cubes, Square anu Cube Roots. 


100 


N umber. 

Squares. 

Cubes. 

| \/ Roots. 

J yj Roots. 

1405 

1974025 

2773505123 

37*4833296 

11-2001913 

1406 

1976836 

2779431416 

37-4966665 

11*2028479 

1407 

1979649 

2785366143 

37-5099987 

11-2055032 

1408 

1982464 

2791309312 

37-5233261 

11-2081573 

1409 

19S52S1 

2797260929 

37-5366487 

11-2108101 

1410 

1988100 

2803221000 

37*5499667 

11-2134617 

1411 

1990921 

2809189531 

37-5632799 

11-2161120 

1412 

1993744 

2815166528 

37-5765885 

11-2187611 

1413 

1996569 

2821151997 

37*5898922 

11-2214089 

1414 

1999396 

2827145944 

37-6031913 

11-2240054 

1415 

2002225 

2833148375 

37-6164857 

11-2267007 

14L6 

2005056 

2839159296 

37-6297754 

11-2293448 

1417 

2007889 

2845178713 

37-6430604 

11-2319876 

1418 

2010724 

2851206632 

37-6563407 

11-2346292 

1419 

2013561 

2857243059 

37-6696164 

11-2372696 

1420 

2016400 

2863288000 

37-682S874 

11-2399087 

1421 

2019241 

2869341461 

37-6961536 

11-2425465 

1422 

2022084 

2S75403448 

37-7094153 

11-2451831 

1423 

2024929 

2S81473967 

37-7226722 

11-2478185 

1424 

2027776 

2887553024 

37-7359245 

11-2504527 

1425 

2030625 

2893640625 

37-7491722 

11*2530856 

1426 

2033476 

2899736776 

37-7624152 

11-2557173 

1427 

2036329 

2905841483 

37*7756535 

11-2583478 

1428 

2039184 

2911954752 

37-7888873 

1 1-2609770 

1429 

2042041 

2918076589 

37-8021163 

11-2636050 

1430 

2044900 

2924207000 

37-8153408 

11-2662318 

1431 

2047761 

2930345991 

37-8285606 

11-2688573 

1432 

2050624 

2936493568 

37-S417759 

11-2714816 

1433 

2053489 

2942649737 

37-8549864 

11-2741047 : 

1434 

2056356 

2948814504 

37-8681924 

11-2767266 

1435 

2059225 

2954987875 

37-8813938 

11-2793472 

1436 

2062096 

2961169856 

37-8945906 

11-2819666 

1437 

2064969 

2967360453 

37*9077828 

11-2845849 

1438 

2067844 

2973559672 

37-9209704 

11-2872019 

1439 

2070721 

2979767519 

37-9341535 

11-2898177 

1440 

2073600 

2985984000 

37-9473319 

11-2924323 

1441 

2076481 

2992209121 

37-9605058 

11-2950457 

1442 

2079364 

2998442888 

37-9736751 

11-2976579 j 

1443 

2082249 

3004685307 

37-9868398 

11-3002688 

1444 

2085136 

3010936384 

38-0000000 

1 1-3028786 

1445 

2088025 

3017196125 

38-0131556 

11-3054871 

1446 

2090916 

3023464536 

38-0263067 

11-3080945 

1447 

2093809 

3029741623 

38-0394532 

11-3107006 

1448 

2096704 

3036027392 

38-0525952 

11-3133056 

1449 ! 

2099601, 

3042321849 

38-0657326 

1 1-3159094 

1450 

2102500 | 

3048625000 

38-0788655 

11*3185119 

1451 

2105401 

3054936851 

38-0919939 

11-3211132 

1452 

2108304! 

3061257408 

38-1051178 

11-3237134 

1453 

2111209 

3067586777 

38-1182371 

11*3263124 

1454 i 

2114116! 

3073924664 

38-1313519 

11-3289102 I 

1455 ; 

2117025 

3080271375 

38-1444622 

11-3315067 

1456 1 

2119936 1 

3086626816 

38-1575681 

11-3341022 1 


Reciprocals. 

•0007117438 
*0007112376 
•0007107321 
•0007102273 
•0007097232 
•0007092199 
•0007087172 
•0007082153 
•0007077141 
•0007072136 
•0007067138 
•0007062147 
•0007057163 
•0007052186 
•0007047216 
•0007042254 
•0007037298 
•0007032349 
•0007027407 
•0007022472 
•0007017544 
•0007012623 
•0007007708 
•0007002801 
•0006997901 
•0006993007 
•0006988120 
•0006983240 
•0006978367 
•0006973501 
•0006968641 
•0006903788 
•0000958942 
•0006954103 
•0006949270 
•0006944444 
•0006939625 
•0006934813 
•0006930007 
•0006925208 
•000692Q415 
•0006915629 
•0006910850 
•0006906078 
•0006901312 
•0006896552 
•0006891799 
•0006S87052 
•0006882312 
•0006877579 
•0006872852 
•0006368132 J 






















110 


Table or Squares, Cubes, Square and Cube Roots. 


Number. 

Squares, j Cubes. 

Roots. 

j Roots. 

1457 

2122849 3092990993 

38-1706693 

11-3366964 

1458 

2125764 30993639121 38-1837662 

11-3392894 

1459 

212S681 3105745579 

38-196S585 

11-3418813 

1460 

2131600 3112136000 

38-2099463 

11-3444719 

1461 

2134521 3118535181 

38*2230297 

11-3470614 

1462 

2137444 3124943128 

38-2361085 

11-3496497 

1463 

2140369 3131359847 

38-2491829 

11-3522368 

1464 

2143296 3137785344 

38-2622529 

11-3548227 

1465 

2146225 3144219625 

38*2753184 

11-3574075 

1466 

2149156 3150662696 

38-2863794 

11-3599911 

1467 

2152089 3157114563 

38-3014360 

11-3625735 

1468 

2155024 3163575232 

38-3144881 

11-3651547 

1469 

2157961 3170044709 

38-3275358 

11-3677347 

1470 

2160900 3176523000 

38-3405790 

11-3703136 

1471 

2163841 3183010111 

38*3536178 

11-3728914 

1472 

2166784 3189506048 

38-3666522 

11-3754679 

1473 

2169729 3196010817 

38-3796821 

11-3780433 

1474 

2172676 3202524424 

38-3927076 

11-3806175 

1475 

2175625 3209046875 

38-4057287 

11-3S31906 

1476 

2178576 3215578176 

38-4187454 

11-3857625 

1477 

2181529 3222118333 

38-4317577 

11-3883332 

1478 

2184484!3228667352 

38-4447656 

11-3909028 

1479 

2187441 3235225239 

38-457769 L 

11-3934712 

1480 

2190400 3241792000 

38-4707681 

11-3960384 

1481 

2193361 3248367641 

38-4837627 

11-3986045 

1482 

2196324 3254952168 

38-4967530 

11-4011695 

1483 

2199289 3261545587 

38-5097390 

11-4037332 

1484 

2202256;3268147904 

38-5227206 

11-4062959 

1485 

2205225 3274759125 

38-5356977 

11-4088574 

I486 

2208196 3281379256 

38-5486705 

11-4114177 

1487 

2211169;3288008303 

38-5616389 

11-4139769 

1488 

2214144 3294646272 

38-5746030 

11-4165349 

1489 

2217121 3301293169 

38-5875627 

11-4190918 

1490 

2220100 3307949000 

38-6005181 

11-4206476 

1491 

2223081;3314613771 

38-6134691 

11-4242022 

1492 

2226064!3321287488 

38-6264158 

11-4267556 

1493 

2229049 3327970157 

38-6393582 

11-4293079 

1494 

2232036 3334661784 

38-6522962 

11-4318591 

1495 

2235025 3341362375 

38-6652299 

11-4344092 

1496 

2238016 3348071936 

38*6781593 

11-4369581 

1497 

2241009 3354790473 

38-6910843 

11-4395059 

1498 

2244004 3361517992 

38-7040050 

11-4420525 

1499 

2247001 3.368254499 

38*7169214 

11-4445980 

1500 

2250000 3375000000 

38-7298335 

11-4471424 

1501 

2253001,3381754501 

38-7427412 

11*4496857 

1502 

2256004 3388518008 

38-7556447 

11-4522278 

1503 

2259009 3395290527 

38-7685439 

11-4547688 

1504 

2262016 3402072064 

38-78143S9 

11-4573087 

1505 

2265025 3408862625 

38-7943294 

11-4598476 

1506 

2268036 341566221.6 

3S-S072158 

11-4623850 

1507 

2271049 3422470843 

38-8200978 

11-4649215 

1508 

227406413429288512 

38-8329757 

11-4674568 


Reciprocals. 

' *0006863412 
*0006858711 
*0006854010 

: *0006849315 
•0006S44627 
*0006839945 
*0006835270 
*0006830601 
*0006825939 
•0006821282 
•0006816633 
•0006811989 
*0006807352 
•0006802721 
*0006798097 
•0006793478 
*0006788866 
•0006784261 
"0006779661 
•0006775068 
•0006770481 
•0006765900 
*0006761325 
*0006756757 
•0006752194 
•0006747638 
*0006743088 
*0006738544 
*0006734007 
•0006729474 
•0006724950 
•0006720430 
•0006715917 
*0006711409 
*0006706908 
•0006702413 
'0006697924 
•0006693440 
•0006688963 
*0006684492 
*0006680027 
*0006675567 
•0006671114 
*0006666667 
*0006662225 
•0006657790 
*0006553360 
•0006648936 
•0006644518 
•0006640106 
•0006635700 
•0006631300 


























Table of Squares, Cubes, Square and Cube Roots. 


Ill 


Number. 

■Squares. 

Cubes. 

y/ Roots. 

1509 

2277081 

3436115229 

38-8458491 

1510 

2280100 

3442951000 

38-8587184 

1511 

2283121 

3449795831 

38-8715834 

1512 

2286144 

3456649728 

38-8844442 

1513 

2289169 

3463512697 

38-8973006 

1514 

2292196 

3470384744 

38-9101529 

1515 

2295225 

3477265875 

38-9230009 

1516 

2298256 

3484156096 

38-9358447 

1517 

2301289 

3491055413 

33-9486841 

1518 

2304324 

3597963832 

38-9615194 

1519 

2307361 

3504881359 

38-9743505 

1520 

2310400 

3511808000 

38-9871774 

1521 

2313441 

3518743761 

39-0000000 

1522 

2316484 

3525688648 

39-0128184 

1523 

2319529 

3532642667 

39-0256326 

1524 

2322576 

3539605824 

39-0384426 

1525 

2325625 

3546578125 

39-0512483 

1526 

2328676 

3553559576 

39-0640499 

1527 

2331729 

3560558183 

39-0768473 

1528 

2334784 

3567549552 

39-0896406 

1529 

2337S41 

3574558889 

39-1024296 

1530 

2340900 

3581577000 

39-1152144 

1531 

2343961 

3588604291 

39-1279951 

153? 

2347024 

3595640768 

39-1407716 

1533 

2350089 

3602686437 

39-1535439 

1534 

2353156 

3609741304 

39-1663120 

1535 

2356225 

3616805375 

39-1790760 

1536 

2359296 

3623878656 

39-1918359 

1537 

2362369 

3630961153 

39-2045915 

1538 

2365444 

3638052872 

39-2173431 

1539 

2368521 

3645153819 

39-2300905 

1540 

2371600 j 

3652264000 

39-2428337 

1541 

2374681 

3657983421 

39-2555728 

1542 

23777641 

3666512088 

39-2683078 

1543 

2380849 

3673650007 

39-2810387 

1544 

2383936 

3680797184 

39-2937654 

1545 

2387025 

3687953625 

39-3064880 

1546 

2390116 

3695119336 

39-3192065 

1547 

2393209 

3702294323 

39*3319208 

1548 

2396304 

3709478592 

39-3446311 

1549 

2399401 

3716672149 

39-3573373 

1550 

2402500 

3723875000 

39-3700394 

1551 

2405601 

3731087151 

39-3827373 

1552 

2408704 

3738308608 

39-3954312 

1553 

2411809 

3745539377 

39-4081210 

1554 

2414916 

3752779464 

39-4208067 

1555 

2418025 

3760028875 

39-4334883 

1556 

2421136, 

3767287616 

39-4461658 

1557 

2424249 j 

3774555693 

39-4588393 

1558 

2427364 

3781833112 

39-4715087 

1559 

2430481 | 

3789119879 

39-4841740 

1560 

2433600 

3796416000 

39-4968353 


yitoots. 

11-4699911 

11-4726242 

11-4750562 

11-4775871 

11-4801169 

11-4826455 

11-4851731 

11-4876995 

11-4902249 

11-4927491 

11-4952722 

11-4977942 

11-5003151 

11-5028348 

11-5053535 

11-5078711 

11-5103876 

11-5129030 

11-5154173 

11-5179305 

11-5204425 

11-5229535 

11-5254634 

11-5279722 

11-5304799 

11-5329865 

11-5354920 

11-5379965 

11-5404998 

11-5430021 

11-5455033 

11-5480034 

11-5505025 

11-5530004 

11-5554972 

11-5579931 

11-5604878 

11-5629815 

11-5654740 

11-5679655 

11-5704559 

11-5729453 

11-5754336 

11-5779208 

11-5804069 

11-5828919 

11-5853759 

11-5878588 

11-5903407 

11-5928215 

11-5953013 

11-5977799 


Reciprocals. 

•0006626905 

•0006622517 

•0006618134 

•0006613757 

•0006609385 

•0006605020 

•0006600660 

•0006596306 

•0006591958 

•0006587615 

•0006583278 

•0006578947 

•00065’74622 

•0006570302 

•0006565988 

•0006561680 

•0006557377 

•0006553080 

•0006548788 

•0006544503 

•0006540222 

•0006535948 

•0006531679 

■0006527415 

•0006523157 

•0006518905 

•0006514658 

•0006510417 

•0006506181 

•0006501951 

•0006497726 

•0006493506 

•0006489293 

•0006485084 

•0006480881 

•0006476684 

•0006472492 

•0006468305 

•0006464124 

•0006459948 

•0006455778 

•0006451613 

•0006447453 

•0006443299 

•0006439150 

•0006435006 

•0006430868 

•0006426735 

•0006422608 

•0006418485 

•0006414368 

•0006410253 























112 


Table of Squares, Cubes, Square and Cube Roots. 








Number 

Squares. 

Cubes. 

\/ Roots. 

Roois. 

Reciprocala. 

1561 

2436721 

3803721481 

39-5094925 

11-6002576 

•0006406150 

1562 

2439844 

3811036328 

39-5221457 

11-6027342 

•0006402049 

1563 

2442969 

3818360547 

39-5347948 

11-6052097 

•0006397953 

1564 

2446096 

3825641444 

39-5474399 

11-6076S41 

•0006393862 

1565 

2449225 

3833037125 

39-5600809 

11-6101575 

•0006389776 

1566 

2452356 

3840389496 

39-5727179 

11-6126299 

•0006385696 

1567 

2455489 

3847751263 

39-5853508 

11-6151012 

•0006381621 

1568 

2458624 

3855123432 

► 39-5979797 

11-6175715 

*0006377551 

1569 

2461761 

3862503009 

39-6106046 

11-6200407 

•0006373486 

1570 

2464900 

3869883000 

39-6232255 

11*6225088 

•0006369427 

1571 

2468041 

3877292411 

39-6358424 

11-6249759 

•0006365372 

1572 

2471184 

3884701248 

39-6484552 

11-6274420 

•0006361323 

1573 

2474329 

3892119157 

39-6610640 

11-6299070 

•0006357279 

1574 

2477476 

3S99547224 

39-6736688 

11-6323710 

•0006353240 

1575 

2480625 

3906984375 

39-6862696 

11-6348339 

•0006349206 

1576 

2483776 

3914430976 

39-6988665 

11-6372957 

•0006345178 

1577 

2486929 

3921887033 

39-7114593 

11-6397566 

•0006341154 

1578 

2490084 

3929352552 

39-7240481 

11-6422164 

•0006337136 

1579 

2493241 

3936827539 

39-7366329 

11-6446751 

•0006333122 

1580 

2496400 

3944312000 

39-7492138 

11-6471329 

•0006329114 

1581 

2499561 

3951805941 

39-7617907 

11-6495895 

•0006325111 

1582 

2502724 

395930936S 

39-7743636 

11-6520452 

•0006321113 

1583 

2505889 

3966822287 

39-7869325 

11-6544998 

•0006317119 

1584 

2509056 

3974344704 

39-7994976 

11-6569534 

•0006313131 

1585 

2512225 

3981876625 

39-8120585 

11-6594059 

•0006309148 

1586 

2515396 

3989418056 

39-8246155 

11-6618574 

•0006305170 

1587 

2518569 

3996969003 

39-8371686 

11-6643079 

•0006301197 

1588 

2521744 

4004529472 

39-8497177 

11-6667574 

•0006297229 

1589 

2524921 

4012099469 

39-S622628 

11-6692058 

•0006293266 

1590 

2528100 

4014679000 

39-8748040 

11-6716532 

•0006289308 

1591 

2531281 

4027268071 

39-8873413 

11-6740996 

•0006285355 

1592 

2534464' 

4034866688 

39-8998747 

11-6765449 

•0006281407 

1593 

2537649 

4042474857 

39-9124041 

11-0789892 

•0006277464 

1594 

2540836 

4050092584 

39-9249295 

11-6814325 

•0006273526 

1595 

2544025 

4057719875 

39-9374511 

11-6838748 

•0006269592 j 

1596 

2547216 

4065356736 

39-94996S7 

11-6863161 

•0006265664 i 

1597 

2550409 

4073003173 

39-9624824 

11-6887563 

•0006261741 

1598 

2553604 

4080659192 

39-9749922 

11-6911955 

•0006257822 

1599 

2556S01 

4088324799 

39-9874980 

11-6936337 

•0006253909 

1600 

2560000 

4096000000 

40-0000000 

11-6960709 

•0006250000 

2V. 

It*. 

X 3 . 

1 /it. 

yX. 

i 

X 

\ 

\/ JV. 

X. 

l/iV3. 


V x. ■ 

\/x 

f'X. 

i */~n*. 

X. 

Vn. 

fiV. 

1 

■$' X 

X-. 

X*. 

X 3 . 

X. 

y^X 3 . 


N 3 . 

A\ 

N9. 

Vm. 

X. 

X2 

1 

1 

1 

1 

J i 

»/'l 

X 3 

X‘ 

iY r 2* 

TV’ 3 * 

\ x' 

\x‘ 

- X. 
























Evolution, 


1.13 


When the number contains Integer and Decimals. 

Example 5. Required the Square Root of 7845*45? In the column of Souares 
you will find, 

+7849-96 = 88-62, +7849-96 = 88-62, 

—7845-45 = 88-52-, —7832-25 = 88-52, 

451 divided by * 1771 = 00-0256. 

y/7845-45 = 88-5256 nearly. 

4®*When the number of ciphers in the integer is even, the number of 
figures taken in the Square column must akO be even ; but when the number 
of figures in the integer is odd, the number taken in the Square column must 
also be odd. 

To find the Cube Root of Numbers exceeding 1600. 

Example 6. Required the Cube Root of 5694958 ? In the Cube column you will 
find, 

+5735339 = 1793 +5735339 = 1793. 

—5694958 = 1783- —5639752 = 1783. 


40381 divided by 


9558 7 = 000-4225, 
^894958 = 178-4225 nearly. 


When the number contains Integer and Decimals. 

Example 7. Required the Cube Root of 4186-586? In the column of Cubes you 
will find, 


+4251-528 = 16-23 
—4186-585 = 16-13- 
64942 


4251-528 = 16-23 
4173-281 = 16-1« 

78247 = 00-083 
$ 4186-586 = 16-183 nearly. 


4®* , The following notice must be particularly attended to, when extracting 
Cube Root of numbers with decimals. 

2 ciphers in the integer must be 5, 8, or 11 ciphers in the Cube column. 

3 “ “ « 3, 6, or 9 “ “ 

4 « « « 4 or 7 “ t( 

5 “ « “ 5’ or 8 “ “ 

6 “ “ “ 6, or 9 « “ 

7 « « «• 7, or 10 “ « 


In the Cube column and 8 


Example 8. Required the Cube Root of 61358*75 ? 
figures you will find, 

+61629-875 = 395* +61629875 = 39-53 

—61358-750 = 3943- —61162984 = 39-43 

271-125 divided by 466891 = 00-05807 

4^61358-75 39-45807. 

To find the Fourth Root. 

Rule. Extract the Square Root of the number as before described, and of 
that root extract the Square Root again, then the last is the Fourth root of 
the number. 

Example 9. Required the fourth root of 2469781 ? 


4 / 2469781 = V 72469781 = yl571-4463 = 39-6467, the answer. 

To find the Sixth Root. 

Rule. Find the Cube Root of the number as before described, and of that 


root extract 
number. 


the Square Root, and then the last is the Sixth root of the 


J 


8 
























Irregular Figures. 


114 


To find the Area and Solidity of Irregular Figures* 

Chapman's rule in the construction of ships, Stockholm, 1775. 



Divide the base A B into any even number of equal parts. 5 = distance between 
the ordinates ; Q — area of the projecting figure. 

Q — —(a -J- 4b -j- 2c -j- 4d -f- 2e -f- 4/ -f- g). . • • 1. 

3 

Suppose this area to revolve around the axis A B and form a solid figure like a 
handle, an urn or a gun; then the solidity C of the figure will be— 

C= 1 L^-{a‘H + 462 2cS -f- 4<f2 + 2e2 + 4/2 + gi). . . .2. 

3 


The practical calculation of these formulas is set up 
as in table for Formula 1. Suppose a — 1.25. 6 = 1.15, 
c = 1.52, d = 1.86, e = 2, /= 1.77, and g = 1.20. 

The distance between the ordinates being 5 = 2, 

o 

then the area will be, Q = —X 28.51 = 19 square of 

3 

whatever measure used. 

The convex surface S of the figure will be, S= 2ir Q 
= 2 X 3.14 X19 = H9.3 square. 


Ordinates. 

Mult. 

Product. 

a 

1.25 

1 

1.25 

6 

1.15 

4 

4.50 

c 

1.52 

2 

3.04 

d 

1.86 

4 

7.44 

e 

2. 

2 

4.00 

f 

1.77 

4 

7.08 

9 

1.20 

1 

1.20 

Q 

9.506 

Vz 

28.51 


This rule can also be employed in calculating the 
cubic contents of earth-work in excavations and 
embankments, in which the ordinates are expressed 
in areas of the sections. 

Suppose a = 36 square feet, yards, metres, or 
whatever unit of measure, 6=30, c = 42, <2=56, 
e = 84, /= 72, and g = 50, the distance between 
the sections being, say, 50 feet. The calculation is 
set up as in the preceding table, namely: 

Volume C= 50 X 323.3 = 1616.5 cubics of what¬ 
ever unit of measure used. 

This rule is universally employed for calculating the areas of water-lines, cross- 
sections and cubic contents of displacement in ships (known as Simpson's rule). 

When the cubic content is required between each section, calculate it as ex¬ 
plained in Excavation and Embankment. 

Surface of Revolution. 

The surface generated by a line revolving around an axis, is equal to the length 
of the line multiplied by the circumference of its centre of gravity. 

N. B. The line, whether straight or curved, must be in the same plane as the 
axis. 

Solidity of Revolution. 

The solidity generated by a plane revolving around an axis, is equal to the area 
Of the plane multiplied by the circumference of its centre of gravity. 

N. B. The revolving plane must be in the same plane as the axis. 


Ordinates. 

Mult. 

Product. 

a 

36 

1 

36 

6 

30 

4 

120 

c 

42 

2 

84 

d 

56 

4 

224 

e 

84 

2 

168 

f 

72 

4 

288 

9 

50 

1 

50 

C 

323.3 


970 
































115 


Table of 8th. Ordinates^ for Railroad Curves* 


Angle. 

W 

1. 7. 

Ordinates. 

a. 6. 1 3. 5. 

4, h. 

Angle. 

W 

1. 7. 

Onh 

3. 6. 

nates . 

3. 5 . 

4 . h. 

l c 

•00084 

•00164 

•00193 

•00218 

53° 

•05313 

•08932 

•11063 

•11773 

2 

•00191 

•00327 

•00409 

•00436 

54 

•05422 

•09130 

•11318 

•12003 

3 

•00299 

•00522 

•00561 

•00659 

5 5 

•05531 

•09308 

•11510 

•12235 

4 

•00382 

•00654 

•00818 

•00S72 

56 

•05646 

•09487 

•11731 

•12466 

5 

•00437 

•00818 

•01023 

•01091 

57 

•05760 

•09673 

•11950 

•12698 

6 

•00573 

•00928 

•01228 

•01309 

58 

•05875 

•09853 

•12170 

•12932 

7 

•00675 

•01173 

•01432 

•01527 

59 

•05989 

•10037 

•12393 

•13162 

8 

•00764 

•01309 

•01639 

•01746 

60 

•06094 

•10220 

•12612 

•13397 

9 

•0 0 845 

•01474 

•01842 

•01964 

61 

•06261 

•10427 

•12840 

•13631 

10 

•00955 

•01637 

•02047 

•021S3 

62 

•06331 

•10593 

•13054 

•13866 

1 1 

•01053 

•01801 

•02250 

•02402 

63 

•06451 

•10781 

•13281 

•14101 

12 

•01146 

•01965 

•02456 

•02620 

64 

•06570 

•10964 

•13505 

•14337 

13 

•01245 

•02129 

•02662 

•02839 

65 

•06681 

•11101 

•13765 

•14573 

14 

•01284 

•02271 

•02861 

•03058 

66 

•06805 

•11342 

•13956 

•14810 

15 

•0143S 

•02461 

•03081 

•03282 

67 

•06914 

•11532 

•14181 

•15048 

1 6 

•01535 

•02625 

•03277 

•03496 

68 

•07040 

•11721 

•14409 

•15286 

1 7 

•01630 

•02789 

•03484 

•03715 

69 

•07168 

•11912 

•14637 

•15526 

18 

•01730 

•02956 

•03693 

•03935 

70 

•07284 

•12103 

•14864 

•15765 

1 9 

•01858 

•03125 

•03996 

•04154 

71 

•07407 

•12294 

•15087 

•16005 

20 

•01922 

•03286 

•04103 

•04374 

72 

•07535 

•12485 

•15323 

•16245 

21 

•02022 

•03453 

•04309 

•04594 

73 

•07656 

•12685 

*15555 

•16487 

22 

•02119 

•03619 

•04522 

•04814 

74 

•07784 

•12877 

•15785 

•16729 

23 

•02215 

•03787 

•04720 

•05034 

75 

•07912 

•13078 

•16016 

•16972 

24 

•02311 

•03934 

•04930 

•05255 

76 

•08040 

•13292 

•16247 

•17216 

25 

•02413 

•04117 

•05138 

•05475 

77 

•08168 

•13472 

•16482 

•17460 

26 

•02508 

•04283 

•05346 

•05696 

78 

•08297 

•13670 

•16716 

•17706 

27 

•02610 

•04457 

•05552 

•05917 

79 

•08426 

•13868 

•16951 

•17951 

28 

•02708 

•04621 

•05761 

•06139 

80 

•08560 

•14070 

•17187 

•18198 

29 

•02813 

•04793 

•05970 

•06361 

81 

•08695 

•14274 

•17423 

•18445 

30 

•02911 

•04970 

•06188 

•06582 

82 

•08829 

•14477 

•17660 

•18694 

31 

•03005 

•05125 

■06386 

•06804 

83 

•08944 

•14681 

•17901 

•18943 

32 

•03107 

•05298 

•06596 

•07027 

84 

•09105 

•14888 

•18140 

•19193 

33 

•03191 

•05464 

•06806 

•07250 

85 

•09235 

•15120 

•18379 

•19444 

34 

•03310 

•05637 

•07016 

•07477 

86 

•09377 

•15304 

•18622 

•19695 

35 

•03412 

•05804 

•07424 

•07695 

87 

•09518 

•15509 

•18865 

•19946 

36 

•03515 

•05992 

•07452 

•07919 

88 

•09660 

•15756 

•19108 

•20201 

37 

•03616 

•06147 

•07646 

•08143 

89 

•09780 

•15931 

•19350 

•20555 

38 

•03718 

•06327 

•07858 

•08367 

90 

•09944 

•16144 

•19597 

•20710 

39 

•03821 

•06492 

•08069 

•08591 

91 

•10098 

•16359 

•19842 

•20966 

40 

•03905 

•06631 

•08243 

•08816 

92 

•10240 

•16575 

•20092 

•21223 

41 

•04030 

•06836 

•08494 

•09041 

93 

•10384 

•16787 

•20338 

•21481 

42 

•04133 

•07012 

•08707 

•09266 

94 

•10537 

•17005 

•20589 

•21740 

4 3 

•04241 

•07182 

•08920 

•09492 

95 

•10692 

•17224 

•20S37 

•22000 

4 4 

•04363 

•07353 

•09130 

•09719 

96 

•10851 

•17444 

•21091 

•22262 

45 

•0,522 

•07531 

•09346 

•09945 

97 

•10997 

•17666 

•21342 

•22523 

46 

•04556 

•07706 

•09562 

•10172 

98 

•11150 

•17888 

•21596 

•22786 

47 

•04682 

•07894 

•09790 

•10400 

99 

•11310 

•18111 

•22800 

•23050 

48 

•04833 

•08059 

•09991 

•10627 

1 00 

•11468 

•18354 

•22107 

•23315 

49 

•04879 

•08236 

•00207 

•10856 

101 

•11626 

•18500 

•22364 

•23596 

50 

•04982 

•08413 

•00422 

•11085 

102 

•11791 

•18793 

•22623 

•23848 

5 1 

•05096 

•08593 

•10639 

•11314 

103 

•11959 

•19021 

•22876 

•24107 

52 

•05204 

•08768 

•10855 

•11543 

104 

•12116 

•19256 

•23147 

•24386 














































116 


Rail Road Curves 


RAIL ROAD CURVES. 

When Railroads are to be connected by curves, we commonly have given the 
distance (chord c,) between the two ends o o of the tracks, and the tangential 
angle t By these the curve is to be constructed. 

Example 1. Fig. 94. The chord C = 168 feet, and the tangential angle 
v = 19° 30'. Required the centre angle w =, and the radius Ii = ? 

w = 2(19° 30') = 39°. R = 33 lc c = 1-4979X168 = 251-647 feet. 

k = See Table for Segments, &c., of a circle. 

By Tangential Angles. 

The curve to be laid out by the three tangential angles ror, ron, and noo , 
each angle = = 6° 30'. Required the chord r — ? 

The centre angle for the chord r is 

2X(6° 30') = 13°, and r = 13 k R = 0-2264X251-647 = 56-974 feet. 

By Angles of Deflexion* 

Divide the centre angle w into an even number of parts = z. Set off at o the 
angle z—ron , and bisect it into ror and ron ,—’find the chord r, and sub-chord 
o, and continue as shown by Figure. 

Example 2. Fig. 94. The tangential angle v = 78°, and the chord C = 638 
feet. Required the centre-angle w = ? Radius R — ? Chord r —? and the sub- 
chord a = ? 

w = 2X78° = 156°. R = 1 **k c — 0-51117X638 = 326-126 feet. 

Let the curve be laid out by 6 angles of deflexion, and z = 4x156° = 26°, and 
r = as* R = 0-4499X326-126 = 146-73 feet, 
a = as* r = 0-4495X146-73 = 66-012 feet. 

By Ordinates. 

Example 3. Fig. 95. The chord C = 368 feet, and v — 36°. Required the 
height h = ? 

h = ^(cosec.-o — cot.-r). 

From.cosec.36° = 1-70130 

Subtract - ------ cot.36° == 1-37638 

The height h = 0-32492X184 = 59-785 feet. 

At x = 92 feet from h. Required the ordinate y ? 


sin .2 = 


2X92 sin.36° 


368 


0-2938926 = sin.l7° 6'. 


y = 1x368^ °^ ^ — cot.36°^= 45-9448 feet. 

By Sul>“Cliords. 

Example 4. Fig. 96. The ends o and o of the tracks form different angles to 
and W to the chord C, and therefore must be connected by two curves of differ¬ 
ent radii, R and r. The chord C — 869 feet, w = 38°, and W— 86°. Required 
the distance from o to the height A, n =? sub-chord b =? sub-chord a —1 
radii R and r = ? 

v = iX38° = 19°, and F= £X86° = 43°. 


869 tan,19° 


tan,19°-)-tan.43 0 


= 234-35 feet. 


b = 234-35 sec.43° = 320-42 feet. 
a = sec,19°(869 — 234-35) = 671-21 ft. 


R = 38 ka = 1-5358X671-21 = 1030-2 ft. 
r = 86* b = 0-73314X320-42 = 234-91 ft. 


By Eight Ordinates. 

Exanple 5. Fig. 100. Required 8 ordinates for a curve of chord (7= 710 feet 
and the centre angle w = 69° ? (See Table on the preceding page.) 

1st and 7th Ordinates 0-07168X710 = 50-8928 feet. 

2nd “ 6th “ 0-11912X710 = 84-5752 “ 

3rd « 5th “ 0-14637X710 = 103-9227 “ 

4th or height h 0-15526X710 = 110-2346 « 










Railroad Curves* 


117 



CO 

*- 

• 


By angles of deflexion. 




tv = 2v, R = w k C — iC cosec. v. 



■ ■■■ 1 .. - . . —■. ■■—■ ■ - ■ , ■ - ■ 

r = z k R, a = z k r = 2r sin. £ 2 . 

1----- ■ -■■■■■ 


0 / C 

7</ 

L ar 


95. 

i?y Ordinates. 
h= iC( cosec.v — cot.v). 
iC ( c °s.^ _ cot , y V 

V sin.t: / 

. 2a? sin.i; 

sin.D =-—— . 

O 


A 


96. By Sub-chords. 

71 = P-fe 1 ^ 1 !..-. /t = ?i tan.F, 

tan.iM-tan. V 

i T7 w = 2v 

i = nsec.F, ^ _ 2r 

a «= sec.»(C — ?i), 

£ 

0 /I 

R 

97. 

Parallel tracks by a reverse curve. 

Formulas same as above. 

The length 0 0 = 2c, length of 
a circle arc l = 0-035h R. 

. 


\ ' Vs N N * N ^ r '\7? f 

99. 

The greatest radius in a reverse 
curve. 

w — i(F+3u), W = w+V — v, 
a = w k R, b == w k R, 

R = C sec.w(sin.F—ysin. 3 F— cos. 9 to). 

0 /1 

IT 


100. 

Ourve by 8 Ordinates. 

The ordinates are calculated in the 
accompanying Table, the chord C =* 1 or 
the unit. 

If the angle w is large, or there he some 
obstacle on the chord C, find the height h 
and lay out the curve by two or more sets 
of 8 ordinates. 

r 6 BVA* 










































118 


By Ordinates and Subchords. 


By Ordinates and Subchords. 

Example 6. Fig. 101. The tangents t being prolonged to where they 
meet at a, divide that angle into two equal parts, say 1V=75°. Required 
the tangents/=! external secant S=1 chords C=1 and the angle w—1 
I Radius of the curve R = 1500 feet. 

t — R cot.75°=1500X0-26794=401-91 feet. 

\ Centre angle w=90—75°=15° for half the curve. 

S=R (sec.l5 J —1) =1500 (1-0352 -1) =52-8 feet. 

The chords C=k R= 0-26104X1500=391-56 feet. 

Measure off’from a the tangents and the external secant. 

Draw the chords C C, and divide them each into eight equal parts. 
In the table of ordinates under w— 15° will be found the 

1st. 7th. 0-01438X391-56=5-631, I 3rd. 6th. 0-03081X391-56=12-063, 

2nd. 6th. 0-02461X391-56=9-636, | 4th. 0-03282X39-56=12-851, 

Thus by only four multiplications, 16 ordinates in the curve is obtained. 

Should there be any obstacles for the chords C. C. as is often the case 
in excavations and on embankments, a line can be drawn further in on 
the track parallel to the chord and the ordinates obtained by subtraction, 
readily understood by the Engineer. 

Ellipse by Ordinates. 

By this arrangement ellipses can be constructed of any proportions. 
One of the two axes is divided into 16 equal parts. The ordinates 
drawn and calculated as shown by the figure 102. 

Parallel Tracks by a Semi-Ellipse, 

Example 7. Fig. 108. The instrument placed at b and b', divide the 
angles W and w each into two equal parts, prolong the chords which 
will meet at a, a point in the curve. Divide the chords each into eight 
equal parts, and draw the ordinates parallel to the tracks as shown in the 
figure. The grand chord C is the unit for calculating the ordinates, 
which latter are alike on both the chords c', c". 

1st 2nd. 3rd. 4th. 6th. 6th. 7th. 

0-1795C 0-2058 C 0-2029C 0-1830C 0T477C 0T091C 0-0586C. 

Suppose the grand chord to be C=2050 feet. 

Required the length of the 6th ordinate? 0-1091X2050=223-655 feet. 

Tracks not Parallel by Elliptic, arc, 

Example 8. Fig. 104. Divide the angles W and w each into two equal 
parts, prolong the subchords until they intersect one another at a , which 
is a point in the curve. Divide the chord C into eight equal parts, join 
a with the 4th division and draw the other ordinates parallel thereto. 

Suppose the angles are W=18° and w=12°, the centre angle will be 30° 
for which the ordinates are to be calculated from the table. The chord 
C=125 feet. Required the 3rd and 5th ordinates 1 0-06188X125=7-335 feet. 

Springing of Rails. 

Example 9. Fig. 105. A rail of L=21 feet is to be curved to a radius 
of R=1250 feet. Required the spring S=1 in sixteenths of an inch. 

24X21 2 

S = - ~ = 8-47 sixteenths. 

1250 

Super Elevation of the External Rail. 

Example 10. Fig. 106. A train running M— 30 miles per hour on a 
curve of R=1550 feet radii, the gauge of the track is G=5 feet. Required 
the angle of inclination v=l and the super elevation of the external 
rail h=l 

30 a 

tan.v =-— = 0-0387=tan. 2° 13'. 

15X1550 

h—G sin.l° 2J'=6X0-02356=0-1178 feet, or nearly 1£ inches. 

It is practically impossible to lay the super elevation to suit the dif¬ 
ferent speeds of trains. If a mean speed is taken, the faster passenger 
trains will wear the outer rail, and the slow or freight train will wear 
the inner rail 













Railroad Curves. 


119 




101 . 

By ordinates and subchords, 
t = R cot. TT=i2 tan. 10 , TF=90— w, 

S = R fsec.tu—1J = R ('cosec. W —1^) 
C=lc R. For k, see table of segments. 


102 . 


Ellipse by ordinates. 

1= 0-4840(7 5= 0-9204(7 

2= 0-6616(7 * 6= 0*9682(7 

3= 0-7808 (7 7= 0-9922(7 

4= 0*8660(7 8 = (7 the unit. 



103. 


Parallel tracks by elliptic curvei 
i=£(7. w= 2v. W= 2 V, 


(7 sin. IT 


(7 sin.w 


2 sin.u 2 sin. V 

See example for ordinates. 



a 


104. 

Tracks not parallel by elliptic arc. 
Angle of the arc = W-\- w. 
Ordinates to he calculated from the table. 


lUo. 


"-—Ii-" 


Spring of Rails. 

„ 1-5L 1 . . . , 

S =— n — = spring m inches. 
R 

2 4 

S— —„—=16ths of an inch. 
R 


106. 



Inclination of tracks in curves; 

tan.u=A=(rsin.v. 
15 R 

Meaning of letters, see example. 










































120 


Laying Out Railway Curves. 


Explanation of the Figures on tlie Following Page. 

The most correct and positive ways of laying out railway curves are by external 
secant or by sinus-versus either to be employed, as the ground permits. The 
operation is well understood by tjie figures 107 and 108. 

The natural secant and sinus-versus are found in the trigonometrical tables. 
Subtract 1 from the natural secant, and the remainder will be the external secant. 
Multiply the external secant by the assumed radius, and the product is the external 
secant s in the same unit of measure as the radius. 

The centre angle is divided by 2 and 2 as many times as may be required for 
setting out the curve. 

Fig. 107 is used when there are obstacles inside the curve, and Fig. 108 when 
the outside is inaccessible. The sinus-versus in the tables, multiplied by the 
assumed radius, will be the height of the curve above the chord. 

When the inside of the curve is obstructed, and the point T of intersection is 
also inaccessible, then the curve can be laid out as illustrated by Fig. 109. 

Fig. 110 illustrates how to lay out a curve by chords of 100 feet. 


Tangential angles for a chord of c —100 feet, and different radii R from 500 feet to 

3 miles (fig. 110). 


R. 

tan. angle. 

O • " 

R. 

tan. angle. 

R. 

tan. angle. 

Feet. 

Feet. 

O 

f 

" 

Miles. 

O 

t 

ft 

500 

5 

43 

46 

3000 

0 

57 

18 

0.125 

4 

20 

26 

600 

4 

46 

29 

3500 

0 

49 

6 

0.25 

2 

10 

13 

700 

4 

5 

33 

4000 

0 

42 

58 

0.5 

1 

5 

6 

800 

3 

34 

52 

4500 

0 

38 

12 

0.75 

0 

43 

25 

900 

3 

10 

59 

5000 

0 

34 

23 

1 mile. 

0 

32 

33 

1000 

2 

51 

53 

5500 

0 

31 

15 

1.25 

. 0 

26 

2 

1100 

2 

36 

16 

6000 

0 

28 

39 

1.5 

0 

21 

42 

1200 

2 

23 

35 

7000 

0 

24 

34 

1.75 

0 

18 

42 

1500 

1 

54 

35 

8000 

0 

21 

30 

2 

0 

16 

17 

2000 

1 

25 

56 

9000 

0 

19 

6 

2% 

0 

13 

1 

2500 

1 

8 

46 

10000 

0 

17 

12 

3 

0 

10 

51 


Fig. 116 illustrates a section of a cut or embankment through sloping ground* 
The meaning of letters is the same as that on the following pages on excavation 
and embankment. 

Fig. 117. Sidings for parallel tracks. —Z> = distance over tangent points; W= 
width between centres of tracks, and R — radius of curvature; v = angle of frog- 
plates. 

The different operations of laying out the curves are so well 
understood by railroad engineers that it is considered unnecessary 
to enter into detailed description. The formulas and figures are 
intended only as a memorandum. 













UAlMttUl) C'tlKVKH. 


131 




107. By external secants. 

External secant s R(hqc.w — 1). 

W - 90 to; to = 90 W, w = 2t>. 
tangent t R cot.W — R tan. to. 

108. liy sinus-versm, 

to ~ 180 — W, c — 2R sin.to. 

Rx*~—, o = 2R ain.v. 
2ain.to 

Rinua-vemia h = R siii.to. 
a; = 90 — 

A 



109. When the point T is inaccessible, 
to = 90 — v, h — 2(i eot.v. 
a -(- d = J2 soc.to. tZ = J6 tan.v. 
a - • /2 aeo.to — 16 tun.v. 



110. Tangential angle for a chord of c — 
100 /ecZ, and different radii R from 
600 feet to 3 miles. 

W - 2v, 8111.1*0:= 

2R 


R 


2 nin.^to 


c 2 R sin. \ to. 



111. Railway cut or embankment through 
side slopes. 

i r , Ti 6 Hiii.(90-|-2« — a) 

6 - - ~r -}-«tan.a. o *= - . ; ; - --/. 

2 8in.(90 — z — a) 

35 = 90-}-*— #. v 90 — * — a. 

6 cos.a 

win.(90 -| -*— a) 
si d(d ain.a sec.* -f r). 



Sidings of parallel tracks. 

$ — WW{R-\W). 
D 


n~~ + W. 


IL 

2 R' 





























122 


Excavation and Embankment, 


EXCAVATION AND EMBANKMENT. 

Example 1. The Road-way of an excavated channel is r = 15 feet, the depth 
D — 9 feet, and the breadth at the top b =»-= 46| feet. Require the slope S — l 

, 46*5 ■ J5 

Formula 6. S — —z -— =* 1*75 or 1$ to 1* 

l X 9 

Example 2. The Road way is to be r = 15, D = 18, and the slope S= If, 
Require the breadth b = ? and the cross-section A = ? 

Formula 4. b = 2 X 18 X 1'25 -f 15 = 60 feet. 

18 / \ 

Formula 7. A = — ^60 + 15 J = 675 square feet. 

Example 3. The Road-way is to be r = 16 feet, the slope S= 1&, and the depth 
D = 11 feet. Required the area of Cross-section A = ? 

Formula 9. A = 11 (11 X If + = 357*5 square feet. 

Example 4. The Road-way r = 18 feet, slope S= ly, d = 14 feet 6 inches, and 
the length from o is l = 55 feet. Required the cubic contents c = ? 

( 14 .K ^ f.OK IQv 

—-[- —)= 11995*676 cubic feet, divided 

by 27 = 444.28 cubic yards. 

Example 5. The Road-way is r = 16 feet, slope S — 1£ feet, D = 17'5, d — 7*4 
and the length L = 100 feet. Required the cubic content C = ? 

Formula 12. C = 10o[l*(— '* + 7 * 4 * + 17 )+ ^(17‘5 +7*4)] 

= 44445 cubic feet, or 1645*4 cubic yards. 

The computation is executed thus. 

17-5 17-5 

7-4 7.4 


* 700 - 24*9 

1225 8 


129*50 199-2 

17*5’ = 306*25 ) From table 
7*4* = 54*76 \ of Squares. 


3) 490-51 ( 163 ; 5 5 | slope, add i 
199-2 ’ 


X 100 = 44445. cubic feet, 















Excavation and Embankment. 


123 



Letters in the Formulas correspond with the Figure. 


= cot. v, 
a = D S, 
a = D cot. v t - 
b = 2 D S + r, 




S- 


2D’ 


1 . 

2 . 

3. 

4. 

5. 

6 . 


A = D(D S + r ) y . 

a. = d(d S + r)y 
dS 


7. 

8 . 

9. 

10 . 


- l *~r + s)’ 1L 


3 ‘ 2 

D a + 


C-i[s( ;i 

+ ^ (D + <*)]. 


12 . 


Letters Denote, 

A and a «= Cross-Sections in square feet, of the excavated channel or 
embankment. 

D and d =-= depth in feet, of the Sections. 

r — width in feet of the Road-Way. 

b = Base in feet of the embankment, or top breadth of the channel. 

L = length in feet, between the two Sections A and a. 

I = length in feet, from the Section a to the point o where the ground is 
level with the road. 

C = cubic contents in feet, between A and a. 

c = cubic contents in feet, between a and o. 

S = slope of the sides. The slope is commonly given in proportions, thus: 
“ Slope = 1£ to 1,” which means, that the side slopes 1£ feet horizontally for 1 
foot vertical. 

v = angle of the slopa 





























































124 


Railroads. 


TEAOTION OK ROADS. 

Letters denote. 

F= tractive force in pound avoir., necessary to overcome the rolling 
friction, and ascending inclined plains. 

M—miles per hour of the train or force F. 

T — weight of the load in tons, including the weight of the carriages. 

On rail-roads T includes the weight of the locomotive and tender. 
t = weight of the locomotive resting on the driving wheels in tons. 
h = vertical rise in feet per 100 of inclined roads. 
b = base in feet per 100 of the inclined road or plain. 
k = tractive coefficient in pound per ton of the load T, as noted in the 
accompanying Table, under the different conditions of the road. 

A — area of one of the two cylinder pistons in a locomotive, in sq. in. 

P — mean pressure of steam in lbs. per sq. in. on cylinder pistons. 

S= stroke of pistons in feet. 

D = diameter of driving wheel in feet. 

H = actual horse power of a locomotive or the power necessary for the 
. load. About 25 percent, is allowed for friction and working pumps. 
f — adherence coefficient of the driving wheels to the rails, in pounds 
per ton of the weight t. 
n = revolutions per minute of driving wheels. 
d — continued working hours of a horse. 

v — velocity in feet per second. V = weight of a horse in pounds. 
Example 11. Fig. 114. The area of one of the two cylinder pistons in 
a locomotive is A =314 square inches, stroke of piston P=2 feet, mean- 
pressure P=80 lbs. per square inch. Driving wheels Z>=4 feet diameter. 
Required the tractive force F=1 of a locomotive. 

F = 314X2X80 = 12560 lbg . the angwer< 

4 

The adhesive force of the driving wheels to the rails,//, must always 
be greater than the retractive force of the. locomotive, otherwise the 
wheels will slip on the track. 

Example 12. Fig. 115. A locomotive of/=15 tons on an inclined plain 
rising = 10 feet, and the base 6=99-5 feet per 100. /= 560, other dimen¬ 
sions being the same as in the preceding example. Required the tractive, 
retractive and adhesive forces 1 


Tractive, 


F _ 314X2X 80 
4 


22-4X15X10=9200 lbs. 


Retractive, F = 22-4X15X10=3360 lbs. 

Adhesive, F = «°X 

100 

Consequently the locomotive can ascend the inclined plain with a 
tractive force of 8358 3360=4998 lbs., without slip in the driving wheels. 

Example 13. Fig. 116. A train of T=200 tons is to be drawn Af=20 
miles per hour on a horizontal track in good condition, k= 4. Required 
retractive force F—l 

F — 200 ( 44 - 1 / 20 ) = 1694-4 lbs. the answer. 

Example 14. Fig. 117. A train of P=150 tons is to be drawn up an in¬ 
clined plain of 6=9 feet in 100, with a speed of M= 16 miles per hour, 
fc=4. Required the necessary horse power of the locomotive H=1 

(22-4X94-44-/16) = 1342-144 horses. 

o iO 

Example 15. Fig. 118. Required the tractive ability F=T of a horse, 
running M= 7 miles per hour, in d —4 continued hours. 

375 

F =—— = 26-8 lbs. the answer. 

7)/4 















Railways and Common Roads. 


125 


a 

114. 

„ AS P 28 M 

V - . n == ) 

D D 

E ASPn ASPM 

11000 376 D 

Adhesive force — ft. 



115. 

• ASP nn . . .. Dn 

p - 22-4th. U-*-, 

Adhesive, ^^>22‘4th. retractive. • 


116. 

1 

F= T (k+ yMj. <ft= Adhesive. I 

B=^+ym, j 

i 


i 

] 

117. 

F=T(22-U+k+*/E). <^=Ad. 

3=~(22-i h+k+yM). 

Slglg 

118. 

F= T (Jc+yMj. t>= 1 *466 M. 

„ 550 375 _. + . , 

F— -- ability of a horse. 

v ■s/d Ms/d 

\ 


119. 

F= T (22.4 h+k-\- s /M). 

F=~^ - ~ M= 0-6821 *. 

v s/d 100 


































Railroads. 


126 


Example 16. n ig. 145. Required, the tractive force F= ? of a load 1 — 5.25 tons 
to be drawn A — 2 miles per hour up a turnpike of h = 9 feet in 100, the road be¬ 
ing newly laid with coarse gravel Jc = h0l 

F= 5.25 (22.4 X 8 + 50 + -|/2) = 1328.30 lbs. 

Suppose a horse to weigh t' = 1000 lbs., working continually in d = 1 hour up 
the turnpike. Required, the tractive ability F— ? per horse? 


F= 


UA - 1000 X 9 = 97.5 lbs. 


V 1 


100 


1328 25 

Number of horses = --— = 13.6, say 14 horses, which will be necessary for 

97.5 

the load under the mentioned circumstances. In these examples it is necessary to 
take J/>1. and d>l. 

. Traction Coefficient at very Slow Speed. ^ 

On railroads in good condition, carriage axles well lubricated, . 4 

On railroads under ordinary, not very good condition, ... 8 

On very smooth stone pavement,.12 

On ordinary street pavements in good condition, . . . .20 

On street pavements and turnpikes,.30 

On turnpikes new laid with coarse gravel and broken stones, . . 50 

On common roads in bad condition,.150 

On natural loose ground or sand.560 

Adherence Coefficient. 


On rails of maximum dryness, . 

“ very dry, . 

“ under ordinary circumstances,. 

“ in wet weather,. 

“ with snow or frost,. 

In railway curves the retractive force is augmented so many per cent, as the 
whole train occupies degrees in the curve. 


/ 

672 

560 

450 

315 

224 


Railway Gauges. 

The most general gauge in coal mines. 

Denver and Rio Grande railway,. 

Rio Grande and Texas,. 

The most general gauge in the United States, England, France, Prus¬ 
sia, Sweden, Mexico, Chili and Peru,. 

The compromised gauge,. 

Camden and Amboy,.. 

In the Southern States and in Russia,.. 

Irish railways,. 

Louisiana and Texas, also in Canada and India, .... 
Great Western in England, . 


Gauge 
feet. in. 


6 

8 * 

9 

10 

3 

6 


Rain-fall in Inches at Different Seasons of the Year. 


Locations. 

Year. 

Spring. 

Summer. 

Fall. 

Winter. 

Nisliny, Taguilsk, Russia, . . 

18.26 

3.35 

9.28 

3.70 

1.93 

Tobolsk, Siberia,. 

17.76 

2.29 

9.05 

4.02 

2.40 

Nertchinsk. Asia,. 

18.13 

2.32 

10.5 

4.96 

0.35 • 

Yakoutsk, East Siberia, . . . 

10.25 

1.46 

3.35 

3.59 

1.85 

Peking, China,. 

23.88 

2.17 

17.7 

3.50 

0.51 

Macao, Quang-tong,. 

67.81 

18.8 

28.0 

17.7 

3.31 

Saigon, India, . 

62.80 

5.86 

28.% 

28.0 

0.04 

Yokohama, Japan,. 

35.02 

7.52 

12.0 

15.2 

0.295 

Manilla, Philip. Islands, . . 

71.31 

4.77 

34.1 

25.6 

4.84 

For rain-fall, see page 359. 



































Navigation. 


127 


TRAVERSE SAILING AND SURVEYING. 

To navigate a vessel upon the supposition that the earth is a level plane, on 
which the meridians are drawn north and south, parallel with each other; and 
the parallels east and west, at right-angles to the former. 

The line TV S represents a meridian north and 
south, the line WE represents a parallel east 
and west. 

A ship in Z sailing in the direction of l V, and 
having reached V, it is required to know her 
position to the point Z, which is measured by the 
line IV, and the angle Nil'-, and imagined by 
the lines l a and a V 

While the vessel is running from l to V, the 
distance is measured by the log and time; and 
the course N Z V is measured by the compass 
commonly expressed in points. 

These four quantities bear the following names. 

d = IV, distance from Z to V in miles. 

C — Nl V, course, or points from the meridian. 

ft = Z a, departure or difference in longitudes, in miles. 

u — at', difference in latitudes, in miles. 

Z = latitude in degrees. 

L — difference in longitude, in degrees or time. 

Traverse Formulas. 



ft = d sin.C, - 

- 

1, 

ft = u tan.C, - 

- 

2, 

ft = 60 cos./ L, 

- 

3, 

ft = %/ d^—u 1 , 


4, 

u — d cos.C, - 

- 

5, 

u = ft cot.C, - 

m 

6 , 

_ 601> cos./ 

U tan. C ’ 

- 

7, 

u = V ^ — ft% 

m 

8, 

A- * 

sin.C’ 

m 

9, 

d= U n - 


10, 

cos. C, 



, 60 L cos./ 

d — —. — ^— 9 

sin. C 

m 

11, 

d = V DHm 2 , 

- 

12, 

cos./ = , * 

- 

13, 

, ^ sin.C 

•“•'■-roar* 

- 

14, 


COS .1 — 


u tan. C 


L = 

L = 


ft 

60cos./ ’ 

d sin. C 
60cos./ ’ 

u tan. C ' 


60cos./ * 

cos .C = ~ , ■ 

Cl 


sin.C =—, 
d 


tan. C = ~, 

u 


sin.C = 
tan. C = 


60-L cos .1 
d ’ 

601/ cos .1 


15, 

16, 

17, 

18, 

19, 

20 , 
21 , 
22 , 
23, 


















128 


Land Surveying. 


Example 1. A vessel sails east-north-east (6 points) 236 miles. Required her 
departure u, and difference in latitude u. 

Formula 1. fr = d sin. C= 236 X sin. 6 points = 218 miles departure, and u — d 
cos. c. = 236 X cos. 6 points = 90.3 miles difference in latitude. 

Example 2. A ship sails in north latitude in a course C= ESE$E = Gf points; 
at a distance of 132 miles she made a difference in longitude of L = 3° 34'. What 
latitude is she in? 

Formula 14. cos. I = d sirK ~ = 132 . X si M * = 0.59832 ; 

60 L 60 X 3 + 34 

or l — 53° 15' the latitude. 

In high latitudes and very long distances, the preceding formulas will not give 
sucli correct results as may be desired, because they are set up with the supposi¬ 
tion that the earth is a level plane; but by the aid of spherical trigonometry we 
are enabled to ascertain courses and distances correctly from and between any 
known points on the earth. (See Spherical Trigonometry.) 


LAND SURVEYING. 

Application of formulas on the preceding page. 



The operation is readily understood by the illustration. When only an azimuth 
compass is used, the course C at each station is measured from the magnetic needle 
or meridian to the direction of the survey. When a theodolite is employed, the 
course C is read as carefully as possible from the compass at the first station, but 
at the second station the angle v between the distances is measured, from which 
subtract the first course, and the remainder will be the second course. At the 
third station subtract the second course from the angle between the distances, 
and the remainder will be the third course, and so on. The calculated course is 
compared with that shown by the compass at each station ; if a difference is ob¬ 
served, there may be some errors in the Subtraction or angle measurement, or 
some local attraction of the magnetic needle, which is sometimes the case near 
great deposits of iron ores. The angles and courses are measured by the theodo¬ 
lite because they cannot be read so delicately on the compass. 

At the 5th station, where the 4th and 6th stations are on the same side of the 
meridian and both north of 5, add the 4th course to the angle 4, 5, 6, and the 
sum is the new course. On return to the 1st station, where the 7th and 2d sta¬ 
tions are both on the same side of the meridian, and one north and the other 
south, add the angle 2, 1,7, to the 7th course, subtract the sum from 180°, and 
the remainder should be the 1st course, which shows the accurtS^ of the suiwey. 

If the measurements are not correct, there will be errors on return to the 
first station, as seen at the foot of the traverse table. The correction for varia¬ 
tion of the compass is made on the map. 












Traverse Table. 


129 


Traverse Table for the Survey. 


Sta- 

Course 

Sin. or cos. 

Dist. 

Latitude. 

Departure. 

tion. 

c. 


c. 


d . 

N. 

s. 

E. 

w. 

1 

N. 35°42 / E., 

1 

cos. 81211 
sin. 5835 


200 

162.42 

® • • 

116.70 

• • • 

2 

S. 63 48 E., 


cos. 4415 
sin. 8972 


185 

• • • 

81.68 

165.98 

• • • 

3 

N. 68 38 E., 


cos. 3643 
sin. 9312 


263 

95.81 

• • • 

244.90 

• • • 

4 

S. 42 25 E., 


' cos. 7382 
sin. 6747 


228 

• • • 

168.31 

153.78 

• • • 

5 

N. 85 51 W., 

4 

cos. 0723 
sin. 9974 


223 

16.12 

• • • 

• • • 

222.42 

6 

S. 72 18 W., 

> 

cos. 3040 
sin. 9526 j 


321 

• • • 

97.58 

• • • 

305.78 

7 

N. 64 27 W., 


f cos. 4313' 
sin. 9022 


170 

73.32 

• • • 

• • • 

153.37 

Sum of N. S. E. and W., 

• 

• 

347.67 

347.57 

681.36 

681.57 

Subtract the smallest 

> • 


• 

347.57 



681.36 

Errors in the measurement, 

• 

• 

0.10 



0.21 


Find the natural sines and cosines in the trigonometrical tables. 

The distance, d , multiplied by the cosine for the course C, will be the difference 
in latitude formula 5. 

The distance, d, multiplied by the sine for the course C, will be the departure 
formula 1. 

The formulas and traverse table will answer for any unit of measure, but if the 
above traverse bad been made in miles, whether on land or sea, each departure 
should be divided by cosine for the mean latitude between each two stations, formula 
16, in order to obtain the true difference in longitude. To divide by cosine is the 
same as to multiply by the secant for the same angle. 


Length of a Degree in Parallel of Latitude. 

Multiply the length of a degree at the equator (60 sea-miles = 69.03 statute 
miles = 110.83 kilometres) by cosine for the latitude, and the product will be the 
length of a degree in parallel of latitude. 

The length of a minute or second at the equator, multiplied by the cosine for 
the latitude, will be the corresponding length in the parallel of that latitude. 


Measurement over Sloping Ground. 


d = 

Sloping dis¬ 
tance. 

b — Base , or hori¬ 
zontal distance. 

h — Difference in 
height. 

v — Angle of the 
slopes. 

d = 

h cosec. v. 

b = d cos. v. 

h — d sin. v. 

h 

sm. v = ~. 
d 

d = 

b sec. v. 

b = h cot. v. 

h, = b tan. v. 

•% 

tan. v = -. 
b 


9 























130 


Mariners’ Compass. 



s 


North. 

South. 

Points. 

Degrees. 

sineC. 

Cos.C. 

tan.C. 


c ( 

JL 

2° 49' 

•0491 

•9988 

•0492 

N. 

s. 4 

a 

5 37 

•0979 

•9952 

•0983 


( 

f 

8 26 

•1544 

•9880 

•1982 

N. by E. 

S. by E. ( 

l 

11 15 

•1936 

•9811 

•1989 

and 

and J 

H 

14 4 

•2430 

•9700 

•2505 

N. by W. 

S. by W. ( 

n 

16 52 

•2901 

•9570 

•3032 



u 

19 41 

•3368 

•9416 

•3577 

N N E 

S. S. E, r 

2 

22 30 

•3827 

•9239 

•4142 


and J 

2* 

25 19 

•4276 

•9039 

•4730 

N N W 

S. S. W. j 

2i 

28 7 

•4713 

•S820 

•5343 



23 

30 56 

•5140 

•8577 

•5993 

N. E. by N. 

S. E. by S. ( 

3 

33 45 

•5555 

•8314 

•6883 

and 

and J 

3 f 

36 44 

•5981 

-8014 

•7463 

N. W. by N. 

s. w. by S. 


39 22 

•6343 

•7731 

•8204 


W \ 

33 

42 11 

•6715 

•7410 

•9062 

\r "P 


4 

45 0 

•707J 

•7071 

1000 

JN. Xi. 

Ps \ 

43 

47 49 

•7410 

•6715 

1103 

ft lid 

and | 

Q IV 1 

43 

50 37 

•7731 

•6345 

1-218 

JN. VV» 

P. VV 

4} 

53 26 

•8014 

•5981 

1-348 

N. E. by E. 

S. E. by E. f 

5 

56 15 

•8314 

•5555 

1-496 

and 

and 4 

53 

59 4 

•8577 

•5140 

1-668 

N. W. by W. 

S. W. by W. ( 

53 

61 52 

•8820 

•4713 

1-870 



53 

64 41 

•9039 

4276 

2114 

E. N. E 

E. S. E. { 

6 

67 30 

•9239 

•3827 

2-414 

and 

and -j 


70 19 

•9416 

•3368 

2-795 

W. N. W. 

W. S. w. ( 

63 

73 7 

•9570 

•2901 

3 295 



63 

75 56 

•9700 

•2430 

3-991 


** , 

7 

78 45 

•9811 

•1936 

5027 

E. by N. 

E. by S. t 

73 

81 34 

•9880 

•1544 

6-744 

and 

and -[ 

73 

84 22 

•9952 

•0979 

1114 

W. by N. 

W. by S. ( 

73 

87 11 

•9988 

•0491 

20 32 

East or 

_ 

West . 

8 

90° | 

1000 

0.000 

1 

00 

- 


































































Navigation. 


131 



Distance and Dip of Horizon, 

from different heights above the surface of the ocean. 


Height. 

Distance. 

Dip. 

Height. 

Distance. 

Dip. 

Height. 

Distance. 

Dip. 

Feet. 

Miles. 

t 

" 

Feet. 

Miles. 

/ 


Feet. 

Miles. 

O' " 

0.582 

1 mile. 

0 

59 

16 

5.29 

3 

56 

150 

16.22 

0 14 07 

1* 

1.31 

0 

59 

17 

5.45 

4 

03 

200 

18.72 

0 16 18 

2 

1.87 

1 

24 

18 

5.61 

4 

11 

300 

22.91 

0 19 56 

3 

2.29 

1 

42 

19 

5.77 

4 

17 

400 

26.46 

0 23 03 

4 

2.63 

1 

58 

20 

5.92 

4 

24 

500 

29.58 

0 25 46 

5 

2.96 

2 

12 

25 

6.61 

4 

55 

1000 

32.41 

0 28 18 

6 

3.24 

2 

25 

30 

7.25 

5 

23 

2000 

59 20 

0 51 42 

7 

3.49 

2 

36 

35 

7.83 

5 

49 

3000 

72.50 

1 3 24 

8 

3.73 

2 

47 

40 

8.37 

6 

14 

4000 

83.70 

1 14 15 

9 

3.96 

2 

57 

45 

8.67 

6 

36 

5000 

93.50 

1 21 54 

10 

4.18 

3 

07 

50 

9.35 

6 

58 

1 mile. 

96.10 

1 24 01 

11 

4.39 

3 

16 

60 

10.25 

7 

37 

11 “ 

1 a 

108.96 

1 35 40 

12 

4.58 

3 

25 

70 

11.07 

8 

14 

2 “ 

123.23 

1 48 20 

13 

4.77 

3 

33 

80 

11.83 

8 

48 

2j “ 

140.64 

2 3 50 

14 

4 95 

3 

41 

90 

12.55 

9 

20 

3 “ 

154.10 

2 15 50 

15 

5.12 

3 

49 

100 

13.23 

9 

51 

5 “ 

199.15 

2 57 15 


* For smaller heights, see Curvature of the Earth. 

The refraction is included in the dip of horizon. 

The distance being the tangent a b in statute miles, at the elevation a c, in feet. 

Example\. The lighthouse at a is 100 feet above the level of the sea. Required 
the distance a b. 

Height 100 feet = 13.23 miles. 

Example 2. The flag of a ship is seen from a in d. Required the distance a d, 
when the flag is known to be 50 feet above the level d' of the sea? 

Height of the light 100 = 13.23 miles a. b, 

Height of the flag 50 = 9.35 “ b d, 

Distance to the ship = 22.58 miles a d. 

Example 3. A steamer is seen at e; the horizon b seen in the masts is assumed 
to be 16 feet above the level e’. Required the distance to the ship? 

Height of the light 100 = 13.23 miles a b , 

The assumed height 16 = 5.29 “ e b, 

Distance to the ship = 7.94 miles a e. 























132 


Curvature of the Earth. 


CORRECTION FOR CURVATURE OF THE EARTH 

IN LEVELING. 

Notation of letters. 

D = distance in miles from the level to the stave or other object, and 
d = the same distance in feet. 

C = correction for curvature in feet at the stave ; always negative, 
c = the same correction in inches. 


C= 


2 D l 


d l 


3486643 


D = 1.2247 V~C. 
d — 1867.3 j/cT 


The accompanying table gives the curvature for distances from 100 feet to 20 
miles. For greater distances see table of Distances and Dip of Horizon. 

Difference of Apparent and True Level or Curvature of the 
Earth, with and without it (“fraction. 


Distance. 

Curvature. 

Curv. and ref. 

Distance. 

Curvature. 

Curv. and ref. 

Feet. 

Inches. 

Feet. 

Miles. 

Feet. 

Feet. 

100 

.0028 

.0002 

1 

0.666 

0.575 

200 

.0115 

.0008 

2 

2.666 

2.283 

300 

.0258 

.0018 

3 

6.000 

5.141 

400 

.0489 

.0033 

4 

10.675 

9.150 

500 

.0717 

.0051 

5 

16.675 

14.291 

600 

.1032 

.0073 

6 

24.083 

20.583 

700 

.1405 

.0100 

7 

32.683 

28.167 

800 

.1835 

.0130 

8 

42.691 

36.591 

900 

. .2223 

.0158 

9 

54.025 

46.031 

1000 

.2868 

.0204 

10 

66.700 

57.175 

1500 

.6453 

.0459 

11 

80.708 

69.175 

2000 

1.147 

.0817 

12 

96.050 

82.325 

2500 

1.792 

.1276 

13 

112.716 

96.616 

3000 

2.581 

.1836 

14 

130.732 

112.058 

3500 

3.513 

.2500 

15 

150.075 

126.633 

4000 

4.589 

.372 

16 

170.750 

147.191 

4500 

5.557 

.396 

17 

192.766 

165.225 

5000 

7.170 

.5110 

18 

216.108 

185.233 

5500 

8.676 . 

.6185 

19 

240.783 

206.391 

6000 

10.324 

.7360 

20 

266.800 

228.683 














Divergency of the Parallel. 


133 


TO FIND THE DIVERGENCY OF THE PARALLEL 

FROM THE PRIME VERTICAL. 

Notation of letters. 

I — latitude of the parallel in degrees. 

v = distance on the prime vertical, expressed in angle of the great circle from 
the base-meridian. 

c = divergency in feet of the parallel at the angle v. 

c — 729000 sin. 2 X l • 

The divergency is calculated in the accompanjdng table for distances from one 
second to one degree, also expressed in feet and miles on the prime vertical. The 
coefficient c = 729000 sin. 2 iv, which, multiplied by the latitude of the parallel in 
degrees, gives the divergency in feet. 

Example 1. Suppose the distance on the prime vertical to be v = & = 6 miles 
and 4770 feet, the latitude of the parallel being 48°. Required the divergency. 

From the table, 0.5551 X 48° = 26.6448 feet, the divergency required. 


Divergency of tire Parallel from the Prime Vertical. 


Distance on prime vertical. 

Coefficient. 

Distance on prime vertical. 

Coefficient. 

Seconds v. 

Feet. 

C. 

Minutes v. 

Miles. 

Feet. 

C. 

1 

101.25 

0.00000434 

1 

1 

795 

0.0154213 

2 

202.5 

0.00001735 

H 

1 

3832.5 

0.0346979 

3 

303.75 

0.00003855 

2 

2 

1590 

0.061685 

4 

405 

0.00006916 

2y 

2 

4627.5 

0.0964 

5 

506.25 

0.0001071 

3 

3 

2585 

0.1387917 

6 

607.5 

0.0001542 

4 

4 

3180 

0.24674 

7 

708.75 

0.0002099 

5 

5 

3975 

0.3855 

8 

810 

0.00027665 

6 

6 

4770 

0.55516 

9 

911.25 

0.0003470 

7 

8 

285 

0.75564 

10 

1012.5 

0.0004284 

8 

9 

1080 

0.98696 

11 

1113.75 

0.00051833 

9 

10 

1875 

1.2491253 

12 

1215 

0.0006168 

10 

11 

2670 

1.5420 

13 

1316.25 

0.00072394 

11 

12 

3465 

1.865820 

14 

1417.50 

0.0008396 

12 

13 

4260 

2.220604 

15 

1518.75 

0.0009638 

13 

14 

5055 

2.6062 

16 

1620 

0.0010966 

14 

16 

570 

3.02256 

17 

1721.25 

0.0012380 

15 

17 

1365 

3.4696 

18 

1822.5 

0.0013879 

16 

18 

2160 

3.94783 

19 

1923.75 

0.0015464 

18 

20 

3750 

4.9965012 

20 

2025 

0.0017135 

20 

23 

60 

6.1680 

25 

2531.25 

0.002677 

25 

28 

4035 

9.637500 

30 

3037.5 

0.0038553 

30 

34 

2730 

13.8785 

35 

3543.75 

0.0052475 

35 

40 

1425 

18.8895 

40 

4050 

0.0068539 

40 

45 

120 

24.6720 

45 

4556.25 

0.0086742 

45 

51 

4095 

31.22815 

50 

5062.5 

0.010709 

50 

57 

2790 

38.5500 

55 

5568.75 

0.012958 

55 

63 

1485 

46.6455 

60 

6075 

0.154213 

60 

69 

180 

55.5151 


The length of minutes and seconds on the parallel is equal to that in the table, 
multiplied by cosine for the latitude. 

These calculations are necessary in running a parallel of latitude by fore and 
back sighting, and also for laying out the parallels and meridians on a map. 
















134 


Trigonometry. 


TRIGONOMETRY. 


Trigonometry is that part, of Geometry which treats of triangles. It is divided 
into two parts—viz., plane and spherical. 

Plane Trigonometry treats of triangles which are drawn (or imagined to be) on 
a plane. Spherical Trigonometry treats of the triangles which are drawn (or 
imagined to be) on a sphere. 

A triangle contains seven quantities—namely, three sides, three angles and the 
surface. When any three of these quantities are given, the four remaining ones 
can by them be ascertained (one side or the area must be one of the given quanti 
ties), and the operation is called solving the triangle, which is only an application 
of arithmetic on geometrical objects. 

For the foundation of the above-mentioned solution, there are assumed eight 
help quantities, which are called Trigonometrical functions, and are here denoted 
with their names and number, corresponding with Figure 126. 

Example 1. Fig. 121. An inclined plane a = 150 feet long, and c = 27 feet, the 
height over its base. What is the angle of inclination C— ? 


Formula 14. 


sin.C — 


27 

150 


0.18000. 


Find 9.18000* in the table of sines, which will be found at 10° 30', which is the 
angle C nearly. 

Example 2. Fig. 122. An oblique-angled triangle has the sides c = 27.6 feet, the 
angle C = 34° 10', and the angle A = 47° 40'. How long is the side a — l 


Formula 1. a — 


c sin.A _ 27.6 X sin. 47° 40' 


sin.C 


sin. 34° 10' 


36.33 feet, the answer. 


By Logarithms. 

log.a = log.c log.sin.A — log.sin.C. 

c + log. 27.6 = 1.44090 

A + log. sin. 47° 40' = 9.86878 

1.30968* 

C — log. sin. 34° 10' = 9,74942 

log. 36.4 = 1.56026, or a = 36.4 feet. 

Example 3. Two ships of war notice a strong firing from a castle. In order to 
be safe, they keep themselves at a distance beyond the reach of the balls from the 
castle. To measure the distance from the castle, they place the vessels 800 yards 
from each other, and observe the angles between the castle and the vessels to be 
A = 63°45', B = 75°50'. What will be the two distances from the castle? 

C = 180 — 63° 45' — 75° 50' = 40° 25'. 

To A the distance will be, 

b = 


sin.C 

To B the distance will be, 
esin.A 
sin.C 


c sin.2? 800 X sin. 75° SO; = Il95 . 75 yards . 


a 


sin. 40° 25' 


800 X sin. 63° 45' 

sin. 40° 25' 


1106.6 yards. 


* The index of a logarithm for a fraction is negative; but in the logarithms for 
the trigonometrical functions, 10 is added to the index, for which it appears so 
much less than 10 as the real negative index. Therefore, when trigonometrical 
logarithms are added, 10’s must be rejected from the sum of the index, which will 
be understood by the examples. 












Trigonometry, 


135 


120 . 



Sinus 

abbreviated 

sin.C. 

Cosinus 

a 

cos.C. 

Sinm-versus 

u 

sinv.C. 

Cosinus-versus 


cosv.C. 

Tangent 

66 

tan.C. 

Cotangent 

66 

cot.C. 

Secant 

66 

sec.C. 

Cosecant 

66 

cosec.C. 


r = Radius of the circle, which is the unit hy which the functions are met- 
sored. 



sin. 2 C+cos. 2 C. 

sec.C 

1 




cos.C’ 

tan.C == 

sin.C 


1 

cos. C 7 

cosec. C 

— sin. C ’ 


1 

sinv.C 

=1 — cos. C, 

tan.C «= 

cot.C’ 

cosv.C 

= 1 — sin. C, 

cot.C = 

cos. C. 

sin.2C 

= 2 sin. C cos. C, 


sin. C ’ 





sin.£C = 

= £/Tsin. 2 C+sinv. s C), 

cot.C = 

1 

tan.C’ 

sin.(C+.5) = sin.C cos. B± 
sin.^cos.C. 


Positive and Negative Signs. 


Angles. 

sin. 

cos. 

sinv. 

CO V. 

tan. 

cot. 

sec. 

cosec. 

+0° 

+0 

+1 

+0 

+1 

+o 

+ GO 

+1 

+oo 

-r90° 

+1 

+0 

+1 

+0 

+oo 

+0 

+0O 

+ 1 

+180° 

+0 

-1 

+2 

+1 

+0 

+C© 

—1 

+00 

+270° 

—1 

+o 

+1 

+2 

+oo 

+' 

+00 

—1 

+360° 

+ 1 

O 

+1 

+0 

+ 1 

+° 

—00 

+1 

-oo 


When a quantity has reached 0 or GO, it has ceased to exist, because it tan 
not he increased or diminished. 

Example. What is the length of the secant for an angle of 74° 18'? 

Secant C = — = 3-695. 


































136 


Right-Angled Twangle. 


FORMULA FOR RIGHT-ANGLED TRIANGLES. 



Say the angle to be C — 60°. In the first column of the table of sines, 60° 
corresponds with 0*86602 in the next column, which is the length of sin. 60°, 
when the radius of the circle is one, or the unit, and the expression sin. 60°X36 
means 0*86602X36 = 31*17672, and likewise with all the other Trigonometrical 
expressions. 

In a triangle the functions for an angle have a certain relation to the oppo¬ 
site side; it is this relationship which enables us to solve the triangle by the ap¬ 
plication of Simple Arithmetic. 

In triangles the sides are denoted by the letters a, b, and c; their respective 
opposite angles are denoted by A, B, and C, and the area by Q. 

Example 1. Fig. 136 The side c in a right angled Triangle being 365 feet, and 
the angle C = 39° 20'. How long is the side a— l 

_ 7 0 c 365 365 

2 - ° = 75TU = m r 39°. 20' “ (163383 = 575 ‘ 86 feet > th ® aMWer ‘ 























Oblique-Angled Triangle. 


137 










































138 


Spherical Trigonometry. 


SPHERICAL TRIGONOMETRY. 

Spherical Trigonometry treats of triangles which are drawn (or im¬ 
agined to be) on the surface of a sphere. Their sides are arcs of the great circle 
of the sphere, and measure by the angle of the arc. Therefore tire trigonometrical 
functions hear quite a different relation to the sides. 

Every section of a sphere cut by a plane is a circle. A line drawn through the 

centre and at right angles to the sectional circle is 
called an axis, and the two points where the axis 
meets the surface of the sphere are called the poles 
of the sectional circle. 

When the cutting plane goes through the centre 
of the sphere, it will pass through the great circle, 
and is then called the Equator for the poles. 
Axis = 'N.S. Equator — G.E. T. W. 

Three great circle-planes, a a'a"a'", b b'b", and 
c c'c", cutting a sphere, NES IT) will form a solid 
angle at the centre O, and a triangle ABC on the 
surface of the sphere, in which the arcs a, b, c, are 
the sides. The angles formed by each two planes 
are congruent to each of the appertinent angles 
A, B and C. 

Spherical Distances. 

For the spherical distances, letters will denote, 
l — lower latitude, in degrees from the equator. 

V — highest latitude, “ “ “ 

C = course, from the latitude l to V. 

C' — course, from “ V to l. 

d — shortest distance between l and V in degrees of the great circle. 

L — difference in longitude between l and Z', in degrees, or time, 
tan. to = cot. V cos .L. 
n — 90 =p l — to. 

— I, when l and V are on one side of the equator. 

-{- l, when Z is on one side, and Z' on the other. Then 

7 sin. Z'cos. n 

cos.cp — « • • • 



sin.C: 


sin. (7 = 


cos .m 

sin.X cos.fr' 

sin .d 1 

sin.X cosi 


1 . 


2 . 


3. 


Example. 

pool. 


sin .d 

Required the shortest distance and course from New York to Liver- 

Z = 40° 42' N. latitude, ) N York 
74°42' W. longitude,/ New * ork ' 

V = 53° 22' N. latitude, 1 T . 

2°62' W. longitude,} Llver P° o1 * 

L — 71° 8' difference in longitude, 
tan. to = cot. 63° 22' X cos. 71° 8' = 13° 31'. 
n = 90° — 13° 31' — 40° 42' = 35° 47'. 


Formula 1. cos. d 


sin. 53° 22'X cos, 35° 47' _ 4?0 g8 , 
cos. 13° 31" 


Shortest distance = 47° X 60 + 58 = 2878 geogi-aphical miles. 

sin.(7 = gi n .- . T l08/ . X C0B - 53 °- 2 - 2 - = 49° 23' = 4| points, 
sin. 47° 58' 

course from New York NEIE. 












Right-Angled Spherical Triangle 


1-39 


BIGHT-ANGLED SPHERICAL TRIANGLE. 



sin.Z* = sin.a sin .B, 
tan.c = tan.a cos .B, 


cot.C = cos.a tan.B, 
tan.c = sin.£ tan.C, 


cos.a = cos .b cos.c, 
cos .B = cos .b sin.C, 


tan.a 


tan.fr 
tan. C' 


sin.c 


tan.fr 
tan. B 7 


sin. a 


sin. fr 
tan. B 7 


sin.C 


cos.B 
cos.fr ’ 


1 , 

2 , 

3, 

4, 

5, 

6 , 

7, 


sin.jE? 


sin.C 


tan.C 


sin.fr 
sin. a’ 

tan.fr 

tan.a 

tan.c 

sin.fr’ 


8 , 


tan.B = 

sm.c 


9 , 


cos .c 


cos. C 
sm7B’ 


10 , 


cos. fr 


cos.B 
sin. C 7 


12 , 

13, 

14, 

15, 

16, 
17, 


cos.c 


cos. a 
cos. fr’ 


11 


, cos.a 


cot.C 
tan .B 7 


18. 


The sum of the three angles in a spherical triangle is greater than two right 
angles, and less than six right angles. 

By Spherical Trigonometry we ascertain distances and courses on the surface 
of the earth ; positions and motions of the heavenly bodies, &c., &c. Examples 
will he furnished in Geography and Astronomy. 


Example, 1. Eig. 140 In a right-angled spherical triangle the side or hypothe- 
nuse a = 36° 20 / , the angle B = 68° 50'. How long is the side b = ? 

Formula 1. sin.6 = sin.a.sin.B = sin.36°20 , Xsin.68°60'. 
a log.sin. 36° 20' = 9:77267 

B log.sin. 68° 50' = 9:96966 

The answer, log.sin. 33° 32'= 9:74233 or 6 = 33° 32'. 



























uo 


Oblique-Angled Spherical Triangle. 


OBLIQUE-ANGLED SPHERICAL TRIANGLE. 

126. 

/b\ 



ay 

\& 



A 





sin.a: sin.i *= sin.JL : sin.Z?, 
sin .6 : sin.c = sin.-B : sin.C, 


sin.£ sin^. 

sin.a =-.—=— , 

sm.B 

sin.c sin.I? 


sin.i = 


sin. C 


t<m.(a+b) - tan-k ' 

tan.(a— b) - tan.c 
tan. 1 {A+B) = cot.U 

tan.4(A-B)-cot.U^^\ 
cot.W = tan.i(B-C,|i^ 
tan.ic = tan.i(a- b) B y 


19 

20 

- 21 , 
22 , 

- 23 , 
24 , 

- 25 , 
26 , 


Example 2. Fig. 141 Oblique angled spherical triangle, c = 72° 30'. B — 17° 3<y. 
C = 79° 50'. 

How long is the side b = ? 

sin.c sin.H sin.72° S0'Xsin.l7° 30' 


Formula 20. sin.6 =-.—~ -no KA /-- 

sin. C sin./9° 50' 

c + log.sin. 72° 30' = 9:97942 

B + log.sin. 17° 30' = 9:47812 

+ = 1:45754 

C + log. sin. 79° 50' = 9: 99312 

The answer log.sin. 16° 50' = 9:46442 or 6= 16° 56.' 

























Oblique Angled Spherical Triangle. 


141 
-1 


OBLIQUE-ANGLED SPHERICAL TRIANGLE. 



tan.4(m+n)tan^(m — n = tan.4(a+c)tan£(a— c) 


tan.m = tan.c cos.A, - 

- 27, 

tan.C = 5 in ' m tan '" 1 . 

sin.(6— m) ’ 

- 28, 

cos. c cos. (b — m) 
cos .a — ' /- 

cos .m 

29, 

„„„ cos.a cos.m 

COS. 71 — - • • - 

cos .c 

- 30, 

b = m+n. 


, cos.c tan. A 

cot.m =» —t-, - - - • 

tan. a 

rH 

CO 

• 

s mm a+b+c S = A+B+C , 


sin.4A - . /ffln.(s —c)sin.(s —tj t . 

V sin .b sm.c 

- 32, 

. • / cos.S cx>s .(S — A) 

sra.ia-A / sin.jB sin.C » 

CO 

CO 

• 


To Find the Area of a Spherical Triangle* 

Let Q he the area of the triangle in square degrees; if JR = radius of the 
sphere, the length of one degree will be, 

2rtR ^ I? 1 

— , or one square degree — 


cot.iQ = 
sin.^Q = 


360 

cot. 4c cot.4a+cos.f? 

sin. B ’ 

sin.Jc sin.4« sin.i? 
cos .\b 


' 3285'58* 


• 1 > 

- 2 , 





























142 


Analytical Geometry. 


ANALYTICAL GEOMETRY AND CONIC SECTIONS. 


riff. 128. 


An equation of a line is generally re¬ 
ferred to rectangular lines, AB = axis 
of ordinate and CD = axis of abscissa 
The position of any point P in the curved 
line P I Q is defined by the rectangular 
distances, y the ordinate and x the abs¬ 
cissa; * and y are variables, depending 
on one another. Any change in either of 
them will produce a change in the other, 
in accordance with the formulae for the 
line. The position of a number of points 
can be determined, located and joined 
into the required line of the equation. 
The ordinate y generally constitutes the first member of the equation, and its 
value is determined by assumed values of the abscissa x. 

The junction of the two axes is called origin, and denoted by o. The line will 
not pass through the origin when the equation has a constant term. 

Properties of Lines Referred to Rectangular Co-ordinates. 



The tangent of any curve, 


The subtangent of any curve, . 


The normal of any curve, 


The subnormal of any curve, . 


• + ■ 1 


HG = y 


dx 

dy' 


PE =^ 1 + %- 

OE=y A . . 

dx 


The point of inflection, I, where convex and 
concave curves tangent, or where a curve re- py 
verses, is when. .— 


dx 2 


= o, or oo. 


Let z denote the length of any curve, then 
The radius of curvature of any curve is 

The ordinate y is a maximum or minimum when 


dz — -j/ dx 2 -{- dy 2 . . 
dz 3 


E = 


dy 

dx 


dx d?y 

0 . . 


. 3. 
4. 

. 5. 

6 . 

. 7. 

8 . 


(See Maxima and Minima.) 

A curve is convex to the axis of abscissa when the ordinate and second differen¬ 
tial coefficient have the same sign, but concave when either of them is positive and 
the other negative I Q is convex, and P I concave, to the abscissa C D. 

A Conic Section is the section obtained when a plane cuts a cone. 

The conic sections are of five different kinds, namely: 

1st. Triangle. When the plane cuts the cone through its axis. 

2d. Circle. When the plane cuts the cone at right angles to its axis. 

3d. Ellipse. When the plane cuts the cone obliquely, passing through the two sides. 
4th. Parabola. When the plane cuts the cone parallel to one side. 

5th. Hyperbola. When the plane cuts the cone at an angle to the axis less than 
the angle of the axis and the side of the cone. 










Hyperbolic Logarithms. 143 


- - -1 

HYPERBOLIC LOGARITHMS. 

Hyperbolic logarithms are used in formulas derived from the calculus when the 
differential cannot be integrated without the aid of hyp. logarithms. The common 
logarithm multiplied by 2.30258509 will be the hyperbolic logarithm, and the hyp. 
log. multiplied by 0.43429448 will be the common logarithm. 

X / 

f* dx x^ 

) — = hyp.log.a5' — hyp.log.x 0 = hyp.log.—. 

X % %0 

0 

Hyperbolic Logarithms. 

No. 

0.0 

0.1 

0.3 

0.3 

0.4: 

0.5 

0.6 

0.7 

0.8 

0.9 

0 

-CO 

-00000 

8.39056 

8.79602 

9.08371 

9.30685 

9.48918 

9.64332 

9.77685 

9.89463 

1 

0.00000 

0.09530 

0.18213 

0.26234 

0.33646 

0.40505 

0.46998 

0.53063 

0.58776 

0.64181 

2 

0.69315 

0.74190 

0.78843 

0.83287 

0.87544 

0.91629 

0.95548 

0.99323 

1.02962 

1.06473 

3 

1.09861 

1.13140 

1.16314 

1.19594 

1.22373 

1.25276 

1.28090 

1.30834 

1.33406 

1.36099 

4 

1.38629 

1.41096 

1.43505 

1.45859 

1.48161 

1.50408 

1.52603 

1.54753 

1.56859 

1.58922 

5 

1.60944 

1.62922 

1.64865 

1.66770 

1.68633 

1.70475 

1.72276 

1.74046 

1.75785 

1.77495 

6 

1.79175 

1.80827 

1.82545 

1.84055 

1.85629 

1.87180 

1.88658 

1.90218 

1.91689 

1.93149 

7 

1.94591 

1.96006 

1.97406 

1.98787 

2.00149 

2.01490 

2.02816 

2.04115 

2.05415 

2.06690 

8 

2 07944 

2.09190 

2.10418 

2.11632 

2.12830 

2.14007 

2.15082 

2.16338 

2.17482 

2.18615 

9 

2.19722 

2.20837 

2.21932 

2.23014 

2.24085 

2.25129 

2.26191 

2.27228 

2.28255 

2.29171 

Hyperbolic Logarithms. 

No. 

0 

1 

3 

3 

L 

5 

6 

7 

8 

9 

10 

2.30258 

2.39589 

2.48491 

2.56494 

2.63906 

2.70805 

2.77259 

2.83321 

2.89037 

2.94444 

20 

.99573 

3.04452 

3.09104 

3.13549 

3.17805 

3.21888 

3.25810 

3.29584 

3.33220 

3.36730 

30 

3.40120 

.43399 

.46574 

.49651 

.52636 

.55535 

.58352 

.61092 

.63759 

.66356 

40 

.68888 

.71357 

.73767 

.76120 

.78419 

.80666 

.82864 

.85015 

.87120 

.89182 

50 

.91202 

.93183 

.95124 

.97029 

.98898 

4.00733 

4.02535 

4.04305 

4.06044 

4.07754 

60 

4.09434 

4.11087 

4.12713 

4.14313 

4.15888 

.17439 

.18965 

.20469 

.21951 

.23411 

70 

.24849 

.26268 

.27667 

.29046 

.30406 

.31749 

.33073 

.34380 

.35671 

.36945 

80 

.38203 

.39445 

.40672 

.41884 

.43082 

.44265 

.45435 

.46591 

.47734 

.48864 

90 

.49981 

.51086 

.52179 

.53260 

.54329 

55388 

.56435 

.57471 

.58497 

.59512 

100 

4.60517 

4.61512 

4.62497 

4.63473 

4.64439 

4.65396 

4.66343 

4.67283 

4 68213 

4.69135 

110 

.70048 

.70953 

.71849 

.72739 

.73619 

.74493 

.75359 

.76217 

.77068 

.77912 

120 

.78749 

.79579 

.80402 

.81218 

.82028 

.82831 

.83628 

.84418 

.85203 

.85981 

130 

.86753 

.87519 

.88280 

.89035 

.89784 

.90527 

.91265 

.91998 

.92725 

.93447 

140 

.94164 

.94876 

.95583 

.96284 

.96981 

.97673 

.98360 

.99043 

.99721 

5.00394 

150 

5.01063 

5.01728 

5.02388 

5.03044 

5.03695 

5.04342 

5.04985 

5.05624 

5.06259 

5.06890 

160 

.07517 

.08140 

.08760 

.09375 

.099S6 

.10594 

.11199 

.11799 

.12396 

.12990 

170 

.13580 

.14166 

.14749 

.15329 

.15905 

.16478 

.17048 

.17615 

.18178 

.18738 

180 

.19295 

.19850 

.20400 

.20948 

.21493 

.22035 

.22574 

.23111 

.23644 

.24175 

190 

.24702 

.25227 

.25750 

.26269 

.26786 

.27300 

.27811 

.28320 

.28826 

.29330 

200 

5.29832 

5.30330 

5.30826 

5.31320 

5.31812 

5.32301 

5.32787 

5.33272 

5.33754 

5.34233 

210 

.34711 

.35186 

.35658 

.36129 

.36597 

.37064 

.37528 

.37989 

.38450 

.38907 

220 

.39363 

.39816 

.40268 

.40717 

.41164 

.41610 

.42053 

.42495 

.42934 

.43372 

230 

.43808 

.44242 

.44674 

.45104 

.45532 

.45958 

.46383 

.46806 

.47227 

.47646 

240 

.48064 

.48479 

.48893 

.49306 

.49716 

.50126 

.50533 

.50939 

.51343 

.51745 

250 

5.52146 

5.52545 

5.52943 

5.53339 

5.53733 

5.54126 

5.54517 

5.54907 

5.55296 

5.55683 

260 

.56068 

.56452 

.56834 

.57215 

.57595 

.57973 

.58349 

.58725 

.59098 

.59471 

270 

• .59842 

.60212 

.60580 

.60947 

.61313 

.61677 

.62040 

.62402 

.62762 

.63121 

280 

.63479 

.63835 

.64191 

.64544. 

.64897 

.65249 

.65599 

.65948 

.66296 

.66642 

290 

.66988 

.67332 

.67675 

.68017 

.68358 

.68697 

.69036 

.69373 

.69709 

.70044 

300 

5.70378 

5.70711 

5.71043 

5.71373 

5.71703 

5.72031 

5.72358 

5.72685 

5.73010 

5.73334 

310 

.73657 

.73979 

.74300 

.74620 

.74939 

.75257 

.75574 

.75890 

.76205 

.76519 

320 

.76832 

.77144 

.77455 

.77765 

.78074 

.78382 

.78690 

.78996 

.79301 

.79606 

330 

.79909 

.80212 

.80513 

.80814 

.81114 

.81413 

.81711 

.82008 

.82304 

.82600 

340 

.82894 

.83188 

.83481 

.83773 

.84064 

.84354 

.84644 

.84932 

.85220 

.85507 

350 

5.85793 

5.86078 

5.86363 

5.86647 

5.86929 

5.87212 5.87493 

5.87773 

5.88053 

5.S8332 






























144 


Conic Sections. 

















































Conic Sections. 


145 
















































TjOfJ An ITJTW8. 


L46 


LOGARITHMS. 

A £i©g aritlim is an exponent of a power to which 10 must be raised to give 
a certain number, which will be understood by this 

£ 


3 

P 

B 

o - 


Oq 

p 

2 . 

<r>- 

P- 

B 


W fa 
£ M 
on »r< 
® O 

c 

0 


log. 100 = 2 because 10 2 = 100. 
log. 10000 = 4 “ 10 4 = 10000 

log. 5012 = 3‘7 “ 103-7 = 5012. 

The unit of the logarithm is called the characteristic or index, and the decimal 
part is called the mantissa, the sum of the characteristic and mantissa is the loga¬ 
rithm. The invariable number 10 is the base for the system of Loga¬ 
rithms. 

It is not necessary that the base should he 10, it can he any number, but nil 
the tables of logarithms now in common use, are calculated with 10 to the 
base. 

The nature of logarithms in connection with their numbers are such, that the 
index cf the logarithm is always one less than the number of figures in the 
number, (when the base of the logarithm is 10,) as, 

index 5012 = 3 
mantissa 5012 = 07 

logarithm 5012 = 37. 

Let 10 be raised to any power x, and 

the power of 10* = ctor log. a = x, 
the power of 10* = 6 or log. b — z. 

Let the product of ah = c and the quotient-^ = c. 


10*X10* = 10*** = ab = c 
10 * b 


or log. c = 
or log. d = x — z. 


a = to * or log. m = zXlog- «• 

\J a — n or log. « = log. a : 3. 

Any number represented by the letters a, b, c, or d, can be a power of 10, which 
expottentis the logarithm for the number. Logarithms are calculated for every 
number with three figures in the accompanying Table, by which any operation in 
Multiplication, Division, Involution and Evolution can be performed by simple 
Addition or Subtraction of Logarithms. Tables of Logarithms are commonly more 
extensive, and calculated for any number of four or five figures, which would 
occupy too much room in this book; but by the proportional parts, the logarithm 
can be found by this Table, to four or five figures. The index of the logarithms 
do not appear in the Table, only the mantissa. It is easily remembered that the 
index is one less, than the number of figures in the number; then when the num¬ 
ber is only one figure, the index is 0 ; and when the number is a fraction, the 
index is negative. 

When the logarithm is to be found for a fraction, we commonly have the 
fraction expressed in a decimal; and then the negative index is equal to the 
number of ciphers before the first figure, and commonly marked after the man¬ 
tissa; thus explained in whole numbers and fractions: 

log. 365 = 2-56229.. log. 0*365 = *56229—1. 
log. 467 = 1-66931 log. 0*0467 = *66931—2. 
log. 7*59 = 0-88024 log. 0*00759 = *88024—3. 

In the accompanying Table of Logarithms, for the trigonometrical lines the 
negative index is marked thus, 

log. sin. 35° 40' = log. 0-5S306 = 1:76572. 














Logarithms. 


147 


To find, the Logarithm of Numbers. 

Example 1. Find the logarithm of 45. 

To 45 in the first column of the Table, answers 65321 in the next oolumn, 
which is the mantissa; index = 1 because 45 is two figures. 

Then, log. 45 = 1*65321, the answer. 

Example 2. Find the logarithm of 768 ? 

Opposite 76 in the first column, answers 88536 in the column marked 8 on the 
top or bottom. Index = 2 because 768 is three figures. 

Then, log. 768 ^ 2*88536. 

Example 3. Find the Logarithm of 6846 ? 

log. 6840 = 3-83505 
Proportional part, 64X0'6 = 384 

log. 6846 = 3*835434 the answer. 

To find the number for a given Logarithm. 

Example 1. What number answers to the logarithm 3-87157 ? 

In the Table you will find in the column of logarithms, that 

log. 7440 = 3-87157. 

Example 2. What number answers to the logarithm 3-801884? 

Given logarithm 3-801884, 

Subt. nearest table log, 3-801400 — log. 6330, 

Divided by proportional part, 69|484|. - 7, 

6337 the req. numb. 

Multiplication by Logarithms. 

Rule. Add together the logarithms of the factors, and the sum is the loga¬ 
rithm of the product. 

Example 1. Multiply 425 by 48. 

To log. 425 = 2*62839, 

Add log. 48 = 1-68124, 

The product, log. 20400 = 4-30963. 

Example 2. Multiply 79600 by 0-435. 

To log. 79600 = 4-90091, 

Add, log. 0-435 = -63848— 1, 

The product log. 34690 = 4-53939. 

Division by Logarithms. 

Rule. From the logarithm of the dividend subtract the logarithm of the di¬ 
visor, and the difference is the logarithm of the quotient. 

Example 1. Divide 43800 by 368. 

From log. 43800 = 4-64147, 

Subtract log. 368 = 2-56584, 

The quotient log. 119 = 2-07563. 

Example 2. Divide 36 by 0.625. 

From log. 36 = 1-55636, 

Subtract, log. 0-625 = *79588-1. 

The quotient, log. 57-6 = 1*76048. 

A negative index follows an opposite operation of its mantissa, as if the man¬ 
tissa is subtracted, add the negative index, and vice versa. 

Envolution by Logarithms. 

Rule. Multiply the logarithm of the number by its exponent, and the pro¬ 
duct is the logarithm of the power of the number. 

Involution by Logarithms. 

Rule. Divide the logarithm of the number by the index of the root, and the 
quotient is the logarithm of the root of the n umber. 




















143 


Logarithms of Numbers. 


I 


No. 100 to 1600. Logarithms. 00000 to 20412. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


43 

100 

00000 

00043 

00087 

00130 

00173 

00217 

00260 

00303 

00346 

00389 

i 

4 

101 

0432 

0475 

0518 

0561 

0604 

0647 

0689 

0732 

0775 

0817 

2 

9 

102 

0860 

0903 

0945 

0988 

1030 

1072 

1115 

1157 

1199 

1242 

3 

13 

103 

1284 

1326 

1368 

1410 

1452 

1494 

1536 

1578 

1620 

1662 

4 

17 

104 

1703 

1745 

1787 

1828 

1870 

1912 

1953 

1995 

2036 

2078 

5 

22 

105 

02119 

02160 

02202 

0224.3 

02284 

02325 

02366 

02407 

02449 

02490 

6 

26 

106 

2531 

2572 

2612 

2653 

2694 

2735 

2776 

2816 

2857 

2898 

7 

30 

107 

2938 

2979 

3019 

3060 

3100 

3141 

3181 

3222 

3262 

3302 

8 

34 

108 

3342 

3383 

3423 

3463 

3503 

3543 

3583 

3023 

3663 

3703 

9 

39 

109 

3743 

3782 

3822 

3862 

3902 

3941 

3981 

4021 

4060 

4100 



110 

04139 

04179 

04218 

04258 

04297 

04336 

04376 

04415 

04454 

04493 



111 

4532 

4571 

4610 

4650 

4689 

4727 

4766 

4805 

4844 

4883 

I 

4 

112 

4922 

4961 

4999 

5038 

5077 

5115 

5154 

5192 

5231 

5269 

L 

o 

113 

5308 

5346 

5385 

5423 

5461 

5500 

5538 

5576 

5614 

5652 

6 

1 L 

114 

5690 

5729 

5767 

5805 

5843 

5881 

5918 

5956 

5994 

6032 

4 

lb 

115 

06070 

06108 

06145 

06183 

06221 

06258 

06296 

06333 

06371 

06408 

0 

21 

116 

6446 

6483 

6521 

6558 

6595 

6633 

6670 

6707 

6744 

6781 

6 

25 

117 

6819 

6856 

6893 

6930 

6967 

7004 

7041 

7078 

7115 

7151 

7 

29 

118 

7188 

7225 

7262 

7298 

7335 

7372 

7408 

7445 

7482 

7518 

8 

33 

119 

7555 

7591 

7628 

7664 

7700 

7737 

7773 

7809 

7846 

7882 

9 

37 

120 

07918 

07954 

07990 

08027 

08063 

08099 

08135 

08171 

08207 

08243 


39 

121 

8279 

8314 

8350 

8386 

8422 

8458 

8493 

8529 

8565 

8600 

i 

4 

122 

8636 

8672 

8707 

8743 

8778 

8814 

8849 

8884 

8920 

8955 

2 

8 

123 

8991 

9026 

9061 

9096 

9132 

9167 

9202 

9237 

9272 

9307 

3 

12 

124 

9342 

9377 

9412 

9447 

9482 

9517 

9552 

9587 

9621 

9656 

4 

16 

125 

09691 

09726 

09760 

09795 

09830 

09864 

09899 

09934 

09968 

10003 

5 

20 

126 

10037 

10072 

10106 

10140 

10175 

10209 

10243 

10278 

10312 

0346 

6 

23 

127 

0380 

0415 

0449 

0483 

0517 

0551 

0585 

0619 

0653 

0687 

7 

27 

128 

0721 

0755 

0789 

0823 

0857 

0890 

0924 

0958 

0992 

1025 

8 

31 

129 

1059 

1093 

1126 

1160 

1193 

1227 

1261 

1294 

1327 

1361 

9 

35 

130 

11394 

11428 

11461 

11494 

11528 

11561 

11594 

11628 

11661 

11694 


37 

131 

1727 

1760 

1793 

1826 

I860 

1893 

1926 

1959 

1992 

2024 


132 

2057 

2090 

2123 

2156 

2189 

2222 

2254 

2287 

2320 

2352 

1 

4 

133 

2385 

2418 

2450 

2483 

2516 

2548 

2581 

2613 

2646 

2678 

2 

7 

134 

2710 

2743 

2775 

2808 

2840 

2872 

2905 

2937 

2969 

3001 

3 

11 

135 

13033 

13066 

13098 

13130 

13162 

13194 

13226 

13258 

13290 

13322 

4 

15 

136 

3354 

3386 

3418 

3450 

3481 

3513 

3545 

3577 

3609 

3640 

5 

19 

137 

3672 

3704 

3735 

3767 

3799 

3830 

3862 

3893 

3925 

3956 

6 

22 

138 

3988 

4019 

4051 

4082 

4114 

4145 

4176 

4208 

4239 

4270 

7 

26 

139 

4301 

4333 

4364 

4395 

4426 

4457 

4489 

4520 

4551 

45S2 

8 

30 

140 

14613 

14644 

14675 

14706 

14737 

14768 

14799 

14829 

14860 

14891 

9 

33 

141 

4922 

4953 

4983 

5014 

5045 

5076 

5106 

5137 

5168 

5198 


35 

142 

5229 

5259 

5290 

5320 

5351 

5381 

5412 

5442 

5473 

5503 

1 

4 

143 

5534 

5564 

5594 

5625 

5655 

5685 

5715 

5746 

5776 

5806 

2 

7 

144 

5836 

5866 

5897 

5927 

5957 

5987 

6017 

6047 

6077 

6107 

3 

11 

145 

16137 

16167 

16197 

16227 

16256 

16286 

16316 

16346 

16376 

16406 

4 

14 

146 

6435 

6465 

6495 

6524 

6554 

6584 

6613 

6643 

6673 

6702 

5 

18 

147 

6732 

6761 

6791 

6820 

6850 

6879 

6909 

6938 

6967 

6997 

6 

21 

148 

7026 

7056 

7085 

7114 

7143 

7173 

7202 

7231 

7260 

7289 

7 

25 

149 

7319 

7348 

7377 

7406 

7435 

7464 

7493 

7522 

7551 

7580 

S 

28 

150 

17609 

17638 

17667 

17696 

17725 

17754 

17782 

17811 

17840 

17869 

9 

32 

151 

7898 

7926 

7955 

7984 

8013 

8041 

8070 

8099 

8127 

8156 



152 

8184 

8213 

8241 

8270 

8298 

8327 

8355 

8384 

8412 

8441 


33 

153 

8469 

8498 

8526 

8554 

8583 

S611 

8639 

8667 

8696 

8724 

1 

3 

154 

8752 

8780 

8808 

8837 

8865 

8893 

8921 

8949 

8977 

9005 

2 

7 

155 

19033 

19061 

19089 

19117 

19145 

19173 

19201 

19229 

19257 

19285 

3 

10 

156 

9312 

9340 

9368 

9396 

9424 

9451 

9479 

9507 

9535 

9562 

4 

13 

157 

9590 

9618 

9645 

9673 

9700 

9728 

9756 

9783 

9811 

9838 

5 

17 

158 

9866 

9893 

9921 

9948 

9976 

20003 

20030 

20058 

20085 

20112 

6 

20 

159 

20140 

20167 

20194 

20222 

20249 

0276 

0303 

0330 

0358 

0385 

7 

o 

23 

NO. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9 

zt> 

30 



































Logarithms op Numbers, 


149 


No. 1600 to 2200. Lo^a rit ums. 20412 to 34242. 


No. 

0 

1 

2 

3 

4 

5 1 

6 

7 1 

8 

9 


31 

160 

20412 

20439 

20466 

20493 

20520 

20548 ! 

20575 

20602 

20G29 

20656 

l 

3 

161 

0683 

0710 

0737 

0763 

0796 

0817 

0844 

0871 

0898 

0925 

2 

6 

162 

0952 

0978 

1005 

1032 

1059 

1085 

1112 

1139 

1165 

1192 

3 

9 

163 

1219 

1215 

1272 

1299 

1325 

1352 

1378 

1405 

1431 

1458 

4 

12 

164 

1484 

1511 

1537 

1564 

1590 

1617 

1643 

1669 

1696 

1722 

5 

16 

165 

21748 

21775 

21801 

21827 

21854 

21SS0 

21906 

21932 

21958 

21985 

6 

19 

166 

2011 

2037 

2063 

2089 

2115 

2141 

2167 

2194 

2220 

2246 

7 

.22 

167 

2272 

2298 

2324 

2350 

2376 

2401 

2427 

2453 

2479 

2505 

8 

25 

168 

2531 

2557 

2583 

2608 

2634 

2660 

2686 

2712 

2737 

2763 

9 

28 

169 

2789 

2814 

2840 

2866 

2891 

2917 

2943 

2968 

2994 

3019 


29 

170 

23045 

23070 

23096 

23121 

2:1147 

23172 

23198 

23223 

23249 

23274 


171 

3300 

3325 

3350 

3376 

3401 

3426 

3452 

3477 

3502 

3528 

1 

O 

172 

3553 

3578 

3603 

3629 

3654 

3679 

3704 

3729 

3754 

3779 

A 

6 

173 

3805 

3830 

3855 

3880 

3905 

3930 

3955 

3980 

4005 

4030 

3 

9 

174 

4055 

4080 

4105 

4130 

4155 

4180 

4204 

4229 

4254 

4279 

4 

12 

175 

24304 

24329 

24353 

24378 

24403 

24428 

24452 

24477 

24502 

24527 

5 

15 

176 

4551 

4576 

4601 

4625 

4650 

4674 

4699 

4724 

4748 

4773 

6 

17 

177 

4797 

4822 

4846 

4871 

4895 

4920 

4944 

4969 

4993 

5018 

7 

20 

178 

5042 

5066 

5091 

5115 

5139 

5164 

5188 

5212 

5237 

5261 

8 

23 

179 

5285 

5310 

5334 

5358 

53S2 

5406 

5431 

5455 

5479 

5503 

9 

26 

180 

25527 

25551 

25575 

25600 

25624 

25648 

25672 

25696 

25720 

25744 


27 

181 

5768 

5792 

5816 

5840 

5864 

5888 

5912 

5935 

5959 

5983 

i 

3 

182 

6007 

6031 

6055 

6079 

6102 

6126 

6150 

6174 

6198 

6221 

2 

5 

183 

6245 

6269 

6293 

6316 

6340 

6364 

6387 

6411 

6435 

6458 

3 

8 

184 

6482 

6505 

6529 

6553 

6576 

6600 

6623 

6647 

6670 

6694 

4 

11 

185 

26717 

26741 

26764 

26788 

26811 

26834 

26858 

26881 

26905 

26928 

5 

14 

186 

6951 

6975 

6998 

7021 

7045 

7068 

7091 

7114 

7138 

7161 

6 

16 

187 

7184 

7207 

7231 

7254 

7277 

7300 

7323 

7346 

7370 

7393 

7 

19 

188 

7416 

7439 

7462 

7485 

7508 

7531 

7554 

7577 

7600 

7623 

8 

22 

189 

7646 

7669 

7692 

7715 

7738 

7761 

7784 

7807 

7830 

7852 

9 

24 

190 

27875 

27898 

27921 

27944 

27967 

27989 

2804 2 

28035 

28058 

28081 



191 

8103 

8126 

8149 

8171 

8194 

8217 

8240 

8262 

8285 

8307 


Ao 

192 

8330 

8353 

8375 

8398 

8421 

8443 

8466 

84S8 

8511 

8533 

I 

o 

193 

8556 

8578 

8601 

8623 

8646 

8668 

8691 

8713 

8735 

8758 

A 

D 

194 

8780 

8803 

8825 

8847 

8870 

8892 

8914 

8937 

8959 

8981 

o 

o 

195 

29003 

29026 

29048 

29070 

29092 

29115 

29137 

29159 

29181 

29203 


1U 

196 

9226 

9248 

9270 

9292 

9314 

9336 

9358 

9380 

9403 

9425 

D 

lo 

197 

9447 

9469 

9491 

9513 

9535 

9557 

9579 

9601 

9623 

9645 

O 

10 

198 

9667 

9688 

9710 

9732 

9754 

9776 

9798 

9820 

9842 

9863 

( 

lo 

199 

9885 

9907 

9929 

9951 

9973 

9994 

30016 

30038 

30060 

30081 

o 

2U 

200 

30103 

30125 

30146 

30168 

30190 

30211 

30233 

30255 

30276 

30298 

y 

Ao 

201 

0320 

0341 

0363 

0384 

0406 

0428 

0449 

0471 

0492 

0514 


23 

202 

0535 

0557 

0578 

0600 

0621 

0643 

0664 

0685 

0707 

0728 

i 

2 

203 

0750 

0771 

0792 

0814 

0835 

0856 

0878 

0899 

0920 

0942 

2 

5 

204 

0963 

0984 

1006 

1027 

1048 

1069 

1091 

1112 

1133 

1154 

3 

7 

205 

31175 

31197 

31218 

31239 

31260 

31281 

31302 

31323 

31345 

31366 

4 

9 

206 

1387 

1408 

1429 

1450 

1471 

1492 

4513 

1534 

1555 

1576 

5 

12 

207 

1597 

1618 

1639 

1660 

1681 

1702 

1723 

1744 

1765 

1785 

6 

14 

208 

1806 

1827 

1848 

1869 

1890 

1911 

1931 

1952 

1973 

1994 

7 

16 

209 

2015 

2035 

2056 

2077 

2098 

2118 

2139 

2160 

2181 

2201 

8 

18 

210 

32222 

32243 

32263 

32284 

32305 

32325 

32346 

32366 

32387 

32408 

9 

21 

211 

2428 

2449 

2469 

2490 

2510 

2531 

2552 

2572 

2593 

2613 



212 

2634 

2654 

2675 

2695 

2715 

2736 

2756 

2777 

2797 

2818 

T 

AM. 

9 

213 

2838 

2858 

2879 

2899 

2919 

2940 

2960 

2980 

3001 

3021 


A 

214 

3041 

3062 

3082 

3102 

3122 

3143 

3163 

3183 

3203 

3224 

A 

ft 

215 

33244 

33261 

33284 

33304 

33325 

33345 

33365 

33385 

33405 

33425 

o 

Q 

216 

3145 

3465 

3486 

3506 

3526 

3546 

3566 

3586 

3606 

3626 


1 1 

217 

3646 

3666 

3686 

3706 

3726 

3746 

3766 

3786 

3806 

3826 


1 Q 

218 

3846 

3866 

3885 

3905 

3925 

3945 

3965 

3985 

4005 

4025 

7 

It) 

IS 

219 

4044 

4064 

4084 

4104 

4124 

4143 

4163 

4183 

4203 

4223 

8 

17 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9 

19 













































50 


Logarithms or Numbers, 


No. 2200 to 2800. Logarithms. 34242 to 44716. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

1 8 

9 


30 

220 

34242 

34262 

34282 

34301 

34321 

34341 

34361 

34380 

34400 

34420 

l 

2 

221 

4439 

4459 

4479 

4498 

4518 

4537 

4557 

4577 

4596 

4616 

2 

4 

222 

4635 

4655 

4674 

4691 

4713 

4733 

4753 

4772 

4792 

4811 

3 

6 

223 

4830 

4850 

4869 

4889 

4908 

4928 

4947 

4967 

4986 

5005 

4 

8 

224 

5925 

5044 

5064 

5083 

5102 

5122 

5141 

5160 

5180 

5199 

5 

to 

225 

35218 

35238 

35257 

35276 

35295 

35315 

35334 

35353 

35372 

35392 

6 

12 

226 

5411 

5430 

5449 

5468 

5488 

5507 

5526 

5545 

5564 

55.83 

7 

14 

227 

5603 

5622 

5641 

5660 

5679 

5698 

5717 

5736 

5755 

5774 

8 

16 

228 

5793 

5813 

5832 

5851 

5870 

5889 

5908 

5927 

6946 

5965 

9 

IS 

229 

230 

5984 

36173 

6003 

36192 

6021 

36211 

6040 

36229 

6059 

36248 

6078 
36267^ 

6097 

36286 

6116 

36305 

6135 

36324 

6154 

36342 


19 

231 

6361 

6380 

6399 

6418 

6436 

6455 

6474 

6493 

6511 

6530 

1 

2 

232 

6519 

6568 

6586 

6605 

6624 

6642 

6661 

6680 

669S 

6717 

2 

4 

233 

6736 

6754 

6773 

6791 

6810 

6829 

6847 

6866 

6884 

6903 

3 

6 

234 

6922 

6940 

6959 

6977 

6996 

7014 

7033 

7051 

7070 

7088 

4 

8 

235 

37107 

37125 

37144 

37162 

37181 

37199 

37218 

37236 

37254 

37273 

5 

10 

236 

7291 

7310 

7328 

7346 

7365 

7383 

7401 

7420 

7438 

7457 

6 

11 

237 

7475 

7493 

7511 

7530 

7548 

7566 

7585 

7603 

7621 

7639 

7 

13 

238 

7658 

7676 

7694 

7712 

7731 

7749 

7767 

7785 

7803 

7822 

8 

15 

239 

7840 

7858 

7876 

7894 

7912 

7931 

7949 

7967 

7985 

8003 

9 

17 

240 

38021 

38039 

38057 

38075 

38033 

38112 

3S130 

3814S 

3S166 

38184 


18 

241 

8202 

8220 

8238 

8256 

8274 

8292 

8310 

8328 

8346 

8364 

1 

2 

242 

8382 

8399 

8417 

8435 

8453 

8471 

8489 

8507 

S525 

8543 

2 

4 

243 

8561 

8578 

8596 

8614 

8632 

8650 

8668 

8686 

8703 

8721 

3 

5 

214 

8739 

8757 

8775 

8792 

8810 

882S 

8846 

8863 

8881 

8899 

4 

7 

245 

38917 

38934 

38952 

38970 

38987 

39005 

39023 

39041 

3905S 

39076 

5 

9 

246 

9094 

9111 

9129 

9146 

9164 

91S2 

9199 

9217 

9235 

9252 

6 

11 

247 

9270 

9287 

9305 

9322 

9340 

9358 

9375 

9393 

9410 

9428 

7 

13 

248 

9445 

9463 

9480 

9498 

9515 

9533 

9550 

9568 

9585 

9602 

8 

14 

249 

9620 

9637 

9655 

9672 

9690 

9707 

9724 

9742 

9759 

9777 

9 

16 

250 

251 

39794 

9967 

39811 

9985 

39829 

40002 

39846 

40019 

39863 

40037 

39S81 

40054 

39898 

40071 

39915 

40U88 

39933 

40106 

39950 

40123 

IT 

252 

40140 

40157 

0175 

0192 

0209 

0226 

0243 

0261 

0278 

0295 

I 

2 

253 

0312 

0329 

0346 

0364 

0381 

0398 

0415 

0432 

0449 

0466 

2 

3 

254 

0483 

0500 

0518 

0535 

0552 

0569 

0586 

0603 

0620 

0637 

O 

5 

255 

40654 

40671 

40688 

40705 

40722 

40739 

40756 

40773 

40790 

40807 

4 

7 

9 

256 

0824 

0841 

0858 

0875 

0892 

0909 

0926 

0943 

0960 

0976 

5 

257 

0993 

1010 

1027 

1044 

1061 

1078 

1095 

1111 

1128 

1145 

6 

10 

25 S 

1162 

1179 

1196 

1212 

1229 

1246 

1263 

1280 

1296 

1313 

7 

12- 

259 

1330 

1347 

1363 

1380 

1397 

1414 

1430 

1447 

1464 

1481 

8 

14 

260 

41497 

41514 

41531 

41547 

41564 

41581 

41597 

41614 

41631 

41647 

9 

15 

261 

1664 

1681 

1697 

1714 

1731 

1747 

1764 

1780 

1797 

1814 


16 

262 

1830 

1847 

1863 

1880 

1896 

1913 

1929 

1946 

1963 

1979 

1 

2 

263 

1996 

2012 

2029 

2045 

2062 

2078 

2095 

2111 

2127 

2144 

2 

3 

264 

2160 

2177 

2193 

2210 

2226 

2243 

2259 

2275 

2292 

2308 

3 

5 

265 

42325 

42341 

42357 

42374 

42390 

42406 

42423 

4243y 

42455 

42472 

4 

6 

266 

2488 

2504 

2521 

2537 

2553 

2570 

2586 

2602 

2619 

2635 

5 

8 

267 

2651 

2667 

2684 

2700 

2716 

2732 

2749 

2765 

2781 

2797 

6 

10 

268 

2813 

2830 

2846 

2862 

2878 

2894 

2911 

2927 

2943 

2959 


11 

269 

2975 

2991 

3008 

3021 

3010 

3056 

3072 

3088 

3104 

3120 

8 

13 

270 

43136 

43152 

43169 

43185 

43201 

43217 

43233 

43249 

43265 

43281 

9 

14 

271 

3297 

3313 

3329 

3345 

3361 

3377 

3393 

3409 

3425 

3441 



272 

3457 

3473 

3489 

3505 

3521 

3537 

3553 

3569 

3584 

3600 


15 

273 

3616 

3632 

3648 

3664 

3680 

3696 

3712 

3727 

3743 

3759 

1 

2 

274 

3775 

3791 

3807 

3823 

3838 

3854 

3870 

3886 

3902 

3917 

2 

3 

275 

43933 

43949 

43965 

43981 

43996 

44012 

44028 

44044 

44059 

44075 

3 

5 

276 

4091 

4107 

4122 

4138 

4154 

4170 

4185 

4201 

4217 

4232 

4 

6 

277 

4248 

4264 

4279 

4295 

4311 

4326 

4342 

4358 

4373 

4389 

5 

8 

278 

4404 

4420 

4436 

4451 

4467 

4483 

4498 

4514 | 

4529 

4545 

G 

9 

279 

4560 

4576 

4592 

4607 

4623 

46 o 8 

4654 

4669 

4685 

4700 

7 

11 

NO. 

0 

i 1 

2 

O 

o 

4 

5 

6 

7 

8 

9 

8 

9 

12 

14 


































1/KiA Kl'l M MM Ol! NflMMI'.KH. 


m 


No. 2800 l,o .MOO. 44710 to 0.5148. 


No. 

0 

1 

2 

Q 

<> 

4 

5 

f> 

7 

8 

0 

in 

280 

44710 

14701 

41717 

44702 

14778 

44708 

44600 

418.21 

41810 

4 18.00 

1 2 

281 

4871 

4880 

4002 

4017 

4032 

4948 

4903 

4079 

4994 

07)10 

2 8 

282 

0020 

6040 

0000 

607! 

67)80 

617/2 

6117 

6188 

6148 

617111 

8 6 

28.! 

0170 

0104 

0200 

6226 

6240 

6266 

6271 

6280 

61101 

6817 

4 0 

281 

0002 

0047 

01102 

61178 

6398 

6408 

64211 

6480 

6464 

6409 

6 8 

28/ 1 

40484 

40000 

40010 

47/6110 

40046 

4///.11 J 

46670 

4.01,01 

1 -,(,(/6 

41,7,21 

01 10 

2 0 

0007 

0002 

001.7 

6082 

0007 

6712 

6726 

6748 

6768 

6778 

7 11 

287 

0788 

0800 

0818 

6H04 

6840 

6804 

6070 

6804 

607/) 

6024 

8 111 

288 

0000 

6064 

(06 1 

6004 

0000 

07)16 

07)37) 

0046 

717)07) 

7/)76 

0 14 

280 

<>000 

0100 

0120 

613. 

0160 

0106 

0180 

0106 

71210 

0226 

—. 

2 Mi 

40240 

40200 

40270 

40286 

40300 

40816 

47 J 88 O 

40846 

4 1 ,103 

471871 


2!) 1 

0880 

0401 

0410 

0434 

18110 

78101 

(470 

7404 

0609 

71628 


202 

0008 

Oi/OO 

0008 

06811 

0608 

01118 

07127 

07142 

7J7167 

717172 

15 

2!).'! 

0087 

0702 

0710 

0731 

0740 

0701 

71770 

71707) 

7187 >6 

71827) 

1 2 

201 

0800 

0800 

08(>4 

0870 

0801 

0000 

0028 

0038 

7,0; ,8 

0907 

2 8 

200 

40082 

40007 

47012 

47020 

47041 

47060 

477)70 

47086 

4 VI 7)7) 

47114 

8 6 

200 

7120 

7114 

7100 

7171! 

7188 

727/2 

7217 

7282 

7240 

7201 

4 71 

2 >7 

7270 

72 ; 10 

7800 

7810 

7334 

7049 

787(8 

7878 

7802 

747)7 

6! 8 

208 

71 !2 

7400 

7401 

7406 

7(80 

74.)4 

757)9 

7624 

7688 

766,1 

0 0 

200 

7007 

7082 

7600 

7011 

7020 

7040 

7064 

7717/0 

708,1 

77/)8 

7 11 

000 

47712 

47727 

47741 

47760 

47770 

47784 

47790 

47 18 

47828 

47842 

8 12 

001 

7807 

7871 

7886 

7000 

7014 

702) 

7048 

7068 

7072 

7980 

0 14 

002 

8001 

8010 

8020 

81)44 

60-78 

87)78 

87)67 

8101 

8110 

8187) 

--— 

000 

8111 

8100 

817'. 

8187 

8202 

8210 

8287) 

8244 

5 269 

0278 


001 

8287 

8002 

8810 

811110 

81814 

8360 

6873 

8887 

8,401 

8410 


m 

48400 

48444 

48108 

484711 

48487 

4 867)1 

486J0 

48537) 

4804 l 

48608 

14 

000 

8072 

8080 

8(101 

8016 

8020 

8748 

r 6-/7 

87171 

87187; 

87<)7) 

ll 1 

007 

8714 

8728 

8742 

8760 

8770 

8766 

8,71/,) 

8,8,18 

8827 

8841 

2 8 

008 

8800 

8800 

8880 

8807 

801) 

8020 

8047/ 

8004 

8068 

8082 

8 4 

000 

8000 

9010 

0</2J 

01)118 

9062 

0(/</i 

07/87) 

9004 

0108 

9122 

4 0 

010 

10100 

40100 

40104 

40178 

40102 

40200 

4022<) 

40281 

4 

4927/2 

6j 7 

01 1 

0270 

0200 

0001 

011J8 

0-5.42 

9340 

9200 

01174 

0888 

047/2 

7; 8 

012 

9416 

0420 

0448 

0467 

04 7 J 

9486 

0499 

00 J 8 

062/ 

0011 

7 10 

.010 

0004 

0008 

0682 

0600 

0010 

0024 

07188 

0001 

007},, 

07179 

8 11 

011 

0000 

0707 

0721 

07114 

0748 

0702 

97 70 

9707) 

98‘,8 

9817 

0 J8 

010 

40801 

49846 

40860 

40872 

40880 

4 00' 2/ 

410)14 

400-27 

40041 

49965 


010 

0000 

0082 

') >00 

60010 

601/24 

67)</87 

67)7)61 

07)7)7,;, 

67)7)79 

07//92 


017 

00100 

00120 

60188 

0147 

0101 

0174 

7/188 

7/27/2 

7/2 J-/ 

7/229 


0(8 

0243 

0260 

0270 

1)284 

1/207 

0811 

7)112;/ 

7/1118 

7/162 

0806 

i/'l 

010 

0070 

0393 

0400 

04*21) 

04.1.1 

7447 

7401 

7474 

7)488 

7/4)1 

1 J 

020 

000 JO 

60020 

60642 

67)060 

60600 

67)688 

60690 

678, JO 

00028 

67)71,17 

2 8 

021 

0001 

0664 

0078 

0001 

1/706 

7/718 

7/7-12 

7/74 „ 

7/7//) 

7/772 

* 

022 

0780 

1)700 

08111 

0620 

0640 

7)8,61} 

7)87/1 

7)8,87) 

08915 

7/8/7 

4 6 

020 

0020 

0934 

0047 

1)1)01 

0074 

7)08,7 

17)7)1 

1014 

17/28 

l‘Al 

6 7 

024 

1066 

1008 

JOHJ 

K)O0 

1108 

1121 

111},/ 

JJ48 

117/2 

JJ76 

0 8 

02/ 

01188 

012<)2 

61216 

61228 

1.1242 

61266 

612718 

61282 

61296 

6187/8 

7 9 

020 

1922 

183)) 

1848 

11012 

1876 

188,8 

147/2 

1 715 

J428 

J441 

8 10 

027 

1406 

1408 

14 1 

1496 

1608 

1021 

1684 

1648 

1601 

1074 

0 J2 

* 

1087 

Jf/01 

1014 

1027 

1040 

1064 

1007 

J 7187) 

10./! 

177/1 



J720 

J 780 

1740 

1760 

17)2 

1780 

1700 

1812 

1326 

1088 


O'/O 

018/1 

0180/7 

61678 

61801 

61001 

61017 

61980 

61048 

610.07 

61970 

12 

Ml 

1980 

J 000 

20<2) 

2022 

‘27)8, > 

2748 

27)01 

27/76 

27)88 

217/1 

] 1 

H ) 

*»• 

2114 

2127 

21 10 

216.1 

21 7.0 

2170 

2102 

227) 0 

2218 

2211 1 

2 2 


2244 

2267 

2270 

2284 

2207 

2810 

2828 

28871 

28/19 

21l</2 

8 4 

*J'/4 

2070 

2088 

2401 

2414 

2427 

2447) 

2468 

2400 

2479 

241/2 

4 6 

oof. 

•»•»•) 

02001 

62617 

62600 

62643 

026; 01 

6267/) 

62682 

,'/2)/9,, 

6207/8 

627,21 

6 0 

Ml) 

2001 

2047 

2001) 

2078 

2080 

27,0 / 

2711 

272/1 

2787 

2, 2) 

6 7 

M> i 

2700 

2770 

2780 

287)2 

281,/ 

2827 

2640 

28,,8 

28071 

2870 

7 8 

Vf’M 

2802 

218 );7 

2017 

2012) 

2948 

2060 

2900 

2082 

2991 

87)7/7 

8 10 

Mif 

0020 

87)8/5 

18)40 

80-/8 

8071 

37>84 

87/,/7 

8110 

.1122 

8186 

9 11 

No. 

0 

1 

2 

0 

O 

4 

5 

6 

7 

8 

9 




































152 


Logarithms of Numbers. 


No. 3400 to 4000. Logarithms. Log. 53148 to 60206. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


13 

340 

53148 

53161 

53173 

53186 

53199 

53212 

53224 

53237 

53250 

53263 

1 

1 

341 

3276 

3288 

3301 

3314 

3326 

3339 

3352 

3364 

3377 

3390 

2 

3 

342 

3403 

3415 

3428 

3441 

3453 

3466 

3479 

3491 

3504 

3517 

3 

4 

343 

3529 

3542 

3555 

3567 

3580 

3593 

3605 

3618 

3631 

3643 

4 

5 

344 

3656 

3668 

3681 

3694 

3706 

3719 

3732 

3744 

3757 

3769 

5 

7 

345 

53782 

53794 

53807 

53820' 

53832 

53845 

53857 

53870 

53882 

53895 

6 

8 

346 

3908 

•3920 

3933 

3945 

3958 

3970 

3983 

3995 

4008 

4020 

7 

9 

347 

4033 

4045 

4058 

4070 

4083 

4095 

4108 

4120 

4133 

4145 

8 

10 

348 

4158 

4170 

4183 

4195 

4208 

4220 

4233 

4245 

4258 

4270 

9 

12 

349 

4283 

4295 

4307 

4320 

4332 

4345 

4357 

4370 

4382 

4394 



350 

54407 

54419 

54432 

54444 

54456 

54469 

54481 

54494 

54506 

54518 



351 

4531 

4543 

4555 

4568 

4580 

4593 

4605 

4617 

4630 

4642 



352 

4654 

4667 

4679 

4691 

4704 

4716 

4728 

4741 

4753 

4765 



353 

4777 

4790 

4802 

4814 

4827 

4839 

4851 

4864 

4876 

4888 



354 

4900 

4913 

4925 

4937 

4949 

4962 

4974 

4986 

4998 

5011 



355 

55023 

55035 

55047 

55060 

55072 

55084 

55096 

55108 

55121 

55133 



356 

5145 

5157 

5169 

5182 

5194 

5206 

5218 

5230 

5242 

5255 


13 

357 

5267 

5279 

5291 

5303 

5315 

5328 

5340 

5352 

■5364 

5376 

1 

1 

358 

5388 

6400 

5413 

5425 

5437 

5449 

5461 

5473 

5485 

5497 

2 

2 

359 

5509 

5522 

5534 

5546 

5558 

5570 

5582 

5594 

5606 

5618 

n 

J 

4 

360 

55630 

55642 

55654 

55666 

55678 

55691 

55703 

55715 

55727 

55739 

4 

5 

361 

5751 

5763 

5775 

5787 

5799 

5811 

5823 

5835 

5847 

5859 

5 

6 

362 

5871 

5883 

5895 

5907 

5919 

5931 

5943 

5955 

5967 

5979 

6 

7 

363 

5991 

6003 

6015 

6027 

6038 

6050 

6062 

6074 

6086 

6098 

7 

8 

364 

6110 

6122 

6134 

6146 

6158 

6170 

6182 

6194 

6205 

6217 

8 

10 

365 

56229 

56241 

56253 

56265 

56277 

56289 

56301 

56312 

56324 

56336 

9 

11 

366 

6348 

6360 

6372 

6384 

6396 

6407 

6419 

6431 

6443 

6455 



367 

6467 

6478 

6490 

6502 

6514 

6526 

6538 

6549 

6561 

6573 



368 

6585 

6597 

6608 

6620 

6632 

6644 

6656 

6667 

6679 

6691 



369 

6703 

6714 

6726 

6738 

6750 

6761 

6773 

6785 

6797 

6808 



370 

56820 

56832 

56844 

56855 

56867 

56879 

56891 

56902 

56914 

56926 



371 

6937 

6949 

6961 

6972 

6984 

6996 

7008 

7019 

7031 

7043 



372 

7054 

7066 

7078 

7089 

7101 

7113 

7124 

7136 

7148 

7159 



373 

7171 

7183 

7194 

7206 

7217 

7229 

7241 

7252 

7264 

7276 


11 

374 

7287 

7299 

7310 

7322 

7334 

7345 

7357 

7368 

7380 

7392 

1 

1 

375 

57403 

57415 

57426 

57438 

57449 

57461 

57473 

57484 

57496 

57507 

2 

2 

376 

7519 

7530 

7542 

7553 

7665 

7576 

7588 

7600 

7611 

7623 

3 

3 

377 

7634 

7646 

7657 

7669 

7680 

7692 

7703 

7715 

7726 

7738 

4 

4 

378 

7749 

7761 

7772 

7784 

7795 

7807 

7818 

7830 

7841 

7852 

5 

6 

379 

7864 

7875 

7887 

7898 

7910 

7921 

7933 

7944 

7955 

7967 

6 

7 

380 

57978 

57990 

58001 

58013 

58024 

58035 

58047 

58058 

58070 

58081 

7 

8 

381 

8092 

8104 

8115 

8127 

8138 

8149 

8161 

8172 

8184 

8195 

8 

9 

382 

8206 

8218 

8229 

8240 

8252 

8263 

8274 

8286 

8297 

8309 

9 

10 

383 

8320 

8331 

8343 

8354 

8365 

8377 

8388 

8399 

8410 

8422 



384 

8433 

8444 

8456 

8467 

8478 

8490 

8501 

8512 

8524 

8535 



385 

58546 

58557 

58569 

58580 

58591 

58602 

58614 

58625 

58636 

58647 



386 

8659 

8670 

8681 

8692 

8704 

8715 

8726 

8737 

8749 

8760 



387 

8771 

8782 

8794 

8805 

8816 

8827 

8838 

8850 

8861 

8872 



388 

8883 

8894 

8906 

8917 

8928 

8939 

8950 

8961 

8973 

8984 



389 

8995 

9006 

9017 

9028 

9040 

9051 

9062 

9073 

9084 

9095 



390 

59106 

59118 

59129 

59140 

59151 

59162 

59173 

59184 

59195 

59207 


10 

391 

9218 

9229 

9240 

9251 

9262 

9273 

9284 

9295 

9306 

9318 

1 

1 

392 

9329 

9340 

9351 

9362 

9373 

9384 

9395 

9406 

9417 

9428 

2 

2 

393 

9439 

9450 

9461 

9472 

9483 

9494 

9506 

9517 

9528 

9539 

3 

3 

394 

9550 

9561 

9572 

9583 

9594 

9605 

9616 

9627 

9638 

9649 

4 

4 

395 

59660 

59671 

59682 

69693 

59704 

59715 

59726 

59737 

59748 

59759 

5 

5 

396 

9770 

9780 

9791 

9802 

9813 

9824 

9S35 

9846 

9857 

9868 

6 

6 

397 

9879 

9890 

9901 

9912 

9923 

9934 

9945 

9956 

9966 

9977 

7 

7 

398 

9988 

9999 

60010 

60021 

60032 

60043 

60054 

60065 

60076 

60086 

8 

8 

399 

60097 

60108 

0119 

0130 

0141 

0152 

0163 

0173 

0184‘ 

0195 

9 

9 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 






























Logarithms of Numbers, 


153 


No. 4000 to 4600. 


Logarithms. Log. 60206 to 66276. 

No. 

0 

1 

2 

3 

4 

5 

1 6 

r~> 

i 

8 

9 


11 

400 

I 60206 

60217 

60228 

60239 

60249 

60260 

60271 

60282 

60293 

60304 

1 

1 

401 

0314 

0325 

0336 

0347 

0358 

0069 

0379 

0390 

0101 

0412 

2 

2 

402 

0423 

0433 

0444 

0455 

0466 

0477 

0487 

0498 

0509 

0520 

3 

3 

403 

0531 

0541 

0552 

0563 

0574 

0584 

0595 

0606 

0617 

0627 

4 

4 

104 

0638 

0649 

0660 

0670 

0681 

0692 

0703 

0713 

0724 

0735 

5 

6 

405 

60746 

60756 

60767 

60778 

60788 

60799 

60810 

60821 

60831 

60842 

6 

7 

406 

0853 

0863 

0874 

0885 

0895 

0906 

0917 

0927 

0938 

0949 

7 

8 

407 

0959 

0970 

0981 

0991 

1002 

1013 

1023 

1034 

1045 

1055 

8 

9 

408 

1066 

1077 

1087 

1098 

1109 

1119 

1130 

1140 

1151 

1162 

9 

10 

409 

1172 

1183 

1194 

1204 

1215 

1225 

1236 

1247 

1257 

1268 



410 

61278 

61289 

61300 

61310 

61321 

61331 

61312 

61352 

61363 

61374 



411 

1384 

1395 

1405 

1416 

1426 

1437 

1448 

1458 

1469 

1479 



412 

1499 

1500 

1511 

1521 

1532 

1542 

1553 

1563 

1574 

1584 



413 

1595 

1606 

1616 

1627 

1637 

1648 

1658 

1669 

1679 

1690 



414 

1700 

1711 

1721 

1731 

1742 

1752 

1763 

1773 

1784 

1794 



415 

61805 

61815 

61826 

61836 

61847 

61857 

61868 

61878 

61888 

61899 



416 

1909 

1920 

1930 

1941 

1951 

1962 

1972 

1982 

1993 

2003 



417 

2014 

2024 

2034 

2045 

2055 

2066 

2076 

2086 

2097 

2107 



418 

2118 

2128 

2138 

2149 

2159 

2170 

2180 

2190 

2201 

2211 



419 

2221 

2232 

2242 

2252 

2263 

2273 

2284 

2294 

2304 

2315 



420 

62325 

62335 

62346 

62356 

62366 

62377 

62387 

62397 

62408 

62418 



421 

2428 

2439 

2449 

2459 

2469 

2480 

2490 

2500 

2511 

2521 



422 

2531 

2542 

2552 

2562 

2572 

2583 

2593 

2603 

2613 

2624 



423 

2634 

2644 

2655 

2665 

2675 

2685 

2696 

2706 

2716 

2726 



424 

2737 

2747 

2757 

2767 

2778 

2788 

2798 

2808 

2818 

2829 



425 

62839 

62849 

62859 

62870 

62880 

62890 

62900 

62910 

62921 

62931 


llF 

426 

2941 

2951 

2961 

2972 

2982 

2992 

3002 

3012 

3022 

3033 

1 

1 

427 

3043 

3053 

3063 

3073 

3083 

3094 

3104 

3114 

3124 

3134 



428 

3144 

3155 

3165 

3175 

3185 

3195 

3205 

3215 

3225 

3236 

t> 

O 

429 

3246 

3256 

3266 

3276 

3286 

3296 

3306 

3317 

3327 

3337 



430 

63347 

63357 

63367 

63377 

63387 

63397 

63407 

63417 

63428 

63438 

D 

0 

431 

3448 

3458 

3468 

3478 

3488 

3498 

3508 

3518 

3528 

3538 


D 

432 

3548 

3558 

3568 

3579 

3589 

3599 

3609 

3619 

3629 

3633 

I 

l 

433 

3649 

3659 

3669 

3679 

3689 

3699 

8709 

3719 

3729 

3739 

o 

o 

434 

3749 

3759 

3769 

3779 

3789 

3799 

8809 

3819 

3829 

3839 



435 

63849 

63859 

63869 

63879 

63889 

63899 

63909 

63919 

63929 

63939 



436 

3949 

3959 

3969 

3979 

3988 

3998 

4008 

4018 

4028 

4038 



437 

4048 

4058 

4068 

4078 

4088 

4098 

4108 

4118 

4128 

4137 



438 

4147 

4157 

4167 

4177 

4187 

4197 

4207 

4217 

4227 

4237 



439 

4246 

4256 

4266 

4276 

4286 

4296 

4306 

4316 

4326 

4335 



440 

64345 

64355 

64365 

64375 

64385 

64395 

64404 

64414 

64424 

64434 



441 

4444 

4454 

4464 

4473 

4483 

4493 

4503 

4513 

4523 

4532 



442 

4542 

4552 

4562 

4572 

4582 

4591 

4601 

4611 

4621 

4631 



443 

4640 

4650 

4660 

4670 

4680 

4689 

4699 

4709 

4719 

4729 



444 

4738 

4748 

4758 

4768 

4777 

4787 

4797 

4807 

4816 

4826 



445 

64836 

64846 

64856 

64865 

64875 

64885 

64895 

64904 

64914 

64924 



446 

4933 

4943 

4953 

4963 

4972 

4982 

4992 

5002 

5011 

5021 



447 

5031 

5040 

5050 

5060 

5070 

5079 

5089 

5099 

5108 

5118 



448 

5128 

5137 

5147 

5157 

5167 

5176 

51S6 

5196 

5205 

5215 



449 

5225 

5234 

5244 

5254 

5263 

5273 

5283 

5292 

5302 

5312 



450 

65321 

65331 

65341 

65350 

65360 

65369 

65379 

65389 

65398 

65408 


9 

451 

5418 

5427 

5437 

5447 

5456 

5466 

5475 

5485 

5495 

5504 

l 

1 

452 

5514 

5523 

5533 

5543 

5552 

5562 

5571 

5581 

5591 

5600 

2 

2 

453 

5610 

5619 

5629 

5639 

5648 

5658 

5667 

5677 

6686 

5696 

O 

3 

454 

5706 

5715 

5725 

5734 

5744 

5753 

5763 

5772 

5782 

5792 

4 

4 

455 

65801 

65811 

65820 

65830 

65839 

65849 

65858 

65868 

65877 

65887 

5 

5 

456 

5896 

5906 

5916 

5925 

5935 

5944 

5954 

5963 

5973 

59S2 

6 

5 

457 

£992 

6001 

6011 

6020 

6030 

6039 

6049 

6058 

6068 

6077 

7 

6 

458 

C087 

6096 

6106 

6115 

6124 

6134 

6143 

6153 

6162 

6172 

8 

7 

459 

6181 

6191 

6200 

6210 

6219 

6229 

6238 

6247 

6257 

6266 

9 

8 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


































154 


Logarithms op Numbers, 


No. 4600 to 5200. Logarithms. Log. 66276 to 71600. 


No. 

o 

1 

2 

3 

4 

5 

1 6 

7 

8 

9 


10 

460 

i66276 

66285 

66295 

66304 

66314 

66323 

66382 

66342 

66351 

66361 

l 

1 

461 

6370 

6380 

6389 

6398 

6408 

6417 

6427 

6436 

6445 

6455 

2 

2 

462 

6464 

6474 

6183 

6492 

6502 

6511 

6521 

6530 

6539 

6549 

3 

3 

463 

6558 

6567 

6577 

6586 

6596 

6605 

6614 

6624 

6633 

6642 

4 

4 

464 

6652 

6661 

6671 

6680 

6689 

6699 

6708 

6717 

6727 

6736 

5 

5 

465 

66745 

66755 

66764 

66773 

66783 

66792 

66801 

66811 

66820 

66829 

6 

6 

466 

6839 

6848 

6857 

6867 

6876 

6885 

6894 

6904 

6913 

C922 

7 

7 

467 

6932 

6941 

6950 

6960 

6969 

6978 

69S7 

6997 

7006 

<015 

8 

8 

468 

7025 

7034 

7043 

7052 

7062 

7071 

7080 

7089 

7099 

7108 

9 

9 

46& 

7117 

7127 

7136 

7145 

7154 

7164 

7173 

7182 

7191 

7201 



470 

67210 

67219 

67228 

67237 

67247 

67256 

67265 

67274 

67284 

67293 



471 

7302 

7311 

7321 

7330 

7339 

7348 

7357 

7367 

7376 

7385 



472 

7394 

7403 

7413 

7422 

7431 

7440 

7449 

7459 

7468 

7477 



473 

7486 

7495 

7504 

7514 

7523 

7532 

7541 

7550 

7560 

7569 



474 

7578 

7587 

7596 

7605 

7614 

7624 

7633 

7642 

7651 

7060 



475 

67669 

67679 

67688 

67697 

67706 

67715 

67724 

67733 

67742 

67752 



476 

7761 

7770 

7779 

7788 

7797 

7806 

7815 

7825 

7834 

7843 



477 

7852 

7861 

7870 

7879 

7888 

7897 

7906 

7910 

7925 

7934 



478 

7943 

7952 

7961 

7970 

7979 

7988 

7997 

8006 

8015 

8024 



479 

8034 

8043 

8052 

8061 

8070 

8079 

808S 

8097 

8106 

8115 



480 

68124 

68133 

68142 

68151 

68160 

68169 

68178 

68187 

68196 

68205 



481 

8215 

8224 

8233 

8242 

8251 

8260 

8269 

8278 

8287 

8296 



482 

8305 

8314 

8323 

8332 

8311 

8350 

8359 

8368 

8377 

8386 



483 

8395 

8404 

8413 

8422 

8431 

8440 

8449 

8458 

8467 

8476 



484 

8485 

8494 

8502 

8511 

8520 

8529 

8538 

8547 

8556 

8565 


Q 

485 

68574 

68583 

6S592 

68601 

68610 

68619 

68628 

68637 

68646 

68655 

I 


486 

8664 

8673 

8681 

8690 

8699 

8708 

8717 

8726 

8735 

8744 

9 

o 

487 

8753 

8762 

8771 

8780 

8789 

8797 

8806 

8S15 

8824 

8833 


9 

488 

8842 

8851 

8860 

8869 

8878 

8886 

8895 

8904 

8913 

8922 

t) 

A 

A 

489 

8931 

8940 

8949 

8958 

8966 

8975 

8984 

8993 

9002 

9011 



490 

69020 

69028 

69037 

69046 

69055 

69064 

69073 

69082 

69090 

69099 

0 


491 

9108 

9117 

9126 

9135 

0144 

9152 

9161 

9170 

9179 

9188 

7 

ft 

492 

9197 

9205 

9214 

9223 

9232 

9241 

9249 

9258 

9267 

9276 


7 

493 

9285 

9294 

9302 

9311 

9320 

9329 

9338 

9346 

9355 

9364 


i 

494 

9373 

9381 

9390 

9399 

9408 

9417 

9425 

9434 

9443 

9452 

if 

o 

495 

69461 

69469 

69478 

69487 

69496 

69504 

69513 

69522 

69531 

69539 



496 

9548 

9557 

9566 

9574 

9583 

9592 

9601 

9609 

9618 

9627 



497 

9636 

9644 

9653 

9662 

9671 

9679 

9688 

9697 

9705 

9714 



498 

9723 

9732 

9740 

9749 

9758 

9767 

9775 

9784 

9793 

9801 



499 

9810 

9819 

9827 

9836 

9S45 

9854 

9862 

9871 

9880 

9888 



500 

69897 

69906 

69914 

69923 

69932 

69940 

69949 

69958 

69966 

69975 



501 

9984 

9992 

70001 

70010 

70018 

70027 

70036 

70044 

70053 

70062 



502 

70070 

70079 

0088 

0096 

0105 

0114 

0122 

0131 

0140 

0148 



503 

0157 

0165 

0174 

0183 

0191 

0200 

0209 

0217 

0226 

0234 



504 

0243 

0252 

0260 

0269 

0278 

0286 

0295 

0303 

0312 

0321 



505 

70329 

70338 

70346 

70355 

70364 

70372 

70381 

70389 

70398 

7(4406 



i 506 

0115 

0424 

0432 

0441 

0449 

0458 

0467 

0475 

0484 

0492 



i 507 

0501 

0509 

0518 

0526 

0535 

0544 

0552 

0561 

0569 

0578 



i 508 

0586 

0595 

0603 

0612 

0621 

0629 

0638 

0646 

0655 

0563 



| 509 

0672 

0680 

0689 

0697 

0706 

0714 

0723 

0731 

0740 

0749 



1 510 

70757 

70766 

70774 

70783 

70791 

70800 

70808 

70817 

70825 

70834 


8 

! 511 

0842 

0851 

0859 

0868 

0876 

0885 

0893 

0902 

0910 

0919 

l 

1 

! 512 

0927 

0935 

0944 

0952 

0961 

0969 

0978 

0986 

0995 

1003 

2 

2 

! 513 

1012 

1020 

1029 

1037 

1046 

1054 

1063 

1071 

1079 

1088 

O 

2 

: 514 

1096 

1105 

1113 

1122 

1130 

1139 

1147 

1165 

1164 

1172 

4 

3 

’ 515 

71181 

71189 

71198 

71206 

71214 

71223 

71231 

71240 

71248 

71267 

0 

4 

516 

1265 

1273 

1282 

1290 

1299 

1307 

1315 

1324 

1332 

1341 

6 

5 

517 

1349 

1357 

1366 

1374 

1383 

1391 

1399 

1408 

1416 

1425 

7 

6 

518 

1433 

3441 

1450 

1458 

1466 

1475 

1483 

1492 

1500 

1508 

8 

6 

519 

1517 

1525 

1533 

1542 

1550 

1559 

1567 

1575 

1584 

1592 

9 

7 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 1 

9 












































Logarithms or Numbers. 


156 


No. 5200 to 08 OO. Logarithms. Log. 71600 to 76343. 


No 

1 0 

1 

2 

3 

1 4 

5 

6 

7 

8 

9 


9 

620 

71600 

71609 

71617 

71625 

71634 

71642 

71650 

71659 

71667 

71675 

l 

1 

521 

1684 

1692 

1700 

1709 

1717 

1725 

1734 

1742 

1750 

1759 

2 

2 

522 

17b7 

1775 

1784 

1792 

1800 

1809 

1817 

1825 

1834 

1842 

3 

3 

523 

1850 

1858 

1867 

1875 

1883 

1892 

1900 

190S 

1917 

1925 

4 

4 

524 

1933 

1941 

1950 

1958 

1966 

1975 

1983 

1991 

1999 

2008 

5 

5 

525 

72016 

72024 

72032 

72041 

72049 

72057 

72066 

72074 

72082 

72090 

6 

5 

526 

2099 

2107 

2115 

2123 

2132 

2140 

2148 

2156 

2165 

2173 

7 

6 

527 

2181 

2189 

2198 

2206 

2214 

2222 

2230 

2239 

2247 

2255 

8 

7 

528 

2263 

2272 

2280 

2288 

2296 

2304 

2313 

2321 

2329 

2337 

9 

8 

529 

2346 

2354 

2362 

2370 

2378 

2387 

2395 

2403 

2411 

2419 



530 

72428 

72436 

72444 

72452 

72469 

72469 

72477 

72485 

72493 

72501 



531 

2509 

2518 

2526 

2534 

2542 

2550 

2558 

2567 

2575 

2583 



532 

2591 

2599 

2607 

2616 

2624 

2632 

2640 

2648 

2656 

2665 



533 

2673 

2681 

2689 

2697 

2705 

2713 

2722 

2730 

2788 

2746 



534 

2751 

2762 

2770 

2779 

2787 

2795 

2803 

2811 

2819 

2827 



535 

72835 

72843 

72852 

72860 

72868 

72876 

72884 

72892 

72900 

72908 



536 

2916 

2925 

2933 

2941 

2949 

2957 

2965 

2973 

2981 

29S9 



537 

2997 

3006 

3014 

3022 

3030 

3038 

3046 

3054 

3062 

3070 



538 

3078 

3086 

3094 

3102 

3111 

3119 

3127 

3135 

3143 

3151 



539 

3159 

3167 

3175 

3183 

3191 

3199 

3207 

3215 

3223 

3231 



540 

73239 

73217 

73255 

73263 

73272 

73280 

73288 

73296 

73304 

73312 



541 

3320 

3328 

3336 

3344 

3352 

3360 

3368 

3376 

3384 

3392 



542 

3100 

3408 

3416 

3424 

3432 

3410 

3448 

3456 

3464 

3472 



543 

3480 

3488 

31.96 

3501 

3512 

3520 

8528 

3536 

3514 

3552 



544 

3560 

3568 

3576 

3584 

3592 

3600 

3608 

3616 

3624 

3632 



545 

73640 

73648 

73656 

73664 

73672 

73679 

73687 

73695 

73703 

73711 



546 

3719 

3727 

3735 

3743 

3751 

3759 

3767 

3775 

3783 

3791 

I 

O 

i 

547 

3799 

3807 

3815 

3823 

3830 

3838 

3846 

3854 

3862 

3870 


A 

548 

3878 

3886 

3894 

3902 

3910 

3918 

3926 

3933 

3941 

3949 

fJ 

1 

A 

549 

3957 

3965 

3973 

3981 

3989 

3997 

4005 

4013 

4020 

4028 



550 

74036 

71044 

74052 

74060 

74068 

74076 

74084 

74092 

74099 

74107 

O 

a. 


551 

4115 

4123 

4131 

4139 

4147 

4155 

4162 

4170 

4178 

4186 


0 

552 

4194 

4202 

4210 

4218 

4225 

4233 

4241 

4219 

4257 

4265 

i 

t> 

553 

4273 

4280 

4288 

4296 

4304 

4312 

4320 

4327 

4335 

4343 

o 

0 

554 

4351 

4359 

4367 

4374 

4382 

4390 

4398 

4406 

4414 

4421 


i 

555 

74429 

74437 

74445 

71453 

74461 

74468 

74476 

74484 

74492 

74500 



556 

4507 

4515 

4523 

4531 

4539 

4547 

4554 

4562 

4570 

4578 



557 

4586 

4593 

4601 

4609 

4617 

4624 

4632 

4640 

4648 

4fi66 



558 

4663 

4671 

4679 

4687 

4695 

4702 

4710 

4718 

4726 

4733 



559 

4741 

4749 

4757 

4764 

4772 

4780 

4788 

4796 

4803 

4811 



560 

74819 

74827 

74834 

74842 

74850 

74858 

74865 

74873 

74881 

74889 



561 

4896 

4904 

4912 

4920 

4927 

4935 

4913 

4950 

4958 

4966 



562 

4974 

4981 

4989 

4997 

5005 

5012 

5020 

5028 

5035 

5043 



563 

5051 

5059 

5066 

5074 

5082 

5089 

5097 

5105 

5113 

5120 



564 

5128 

5136 

5143 

5151 

5159 

5166 

5174 

5182 

5189 

5197 



565 

75205 

75213 

75220 

75228 

75236 

75243 

75251 

75259 

75266 

75274 



566 

5282 

5289 

5297 

5305 

5312 

5320 

5328 

5335 

5343 

5351 



567 

5358 

5366 

5374 

5381 

5389 

5397 

5404 

5412 

5420 

5427 



568 

5435 

5442 

5150 

5458 

5465 

5473 

5481 

5488 

5496 

5504 



569 

5511 

5519 

5526 

5534 

5542 

5549 

5557 

5565 

5572 

5580 



570 

75587 

75595 

75603 

75610 

75618 

75626 

75633 

75641 

75648 

75656 


7 

571 

5664 

5671 

5679 

5686 

5694 

5702 

5709 

5717 

5724 

5732 

i 

1 

572 

5710 

5747 

5755 

5762 

5770 

5778 

5785 

5793 

5800 

5808 

2 

1 

573 

5815 

5823 

5831 

5838 

5846 

5853 

5861 

5868 

5876 

5884 

O 

2 

574 

5S91 

5899 

5906 

5914 

5921 

5929 

5937 

5944 

5952 

5959 

4 

3 

575 

75967 

75974 

75982 

75989 

75997 

76005 

76012 

76020 

76027 

76035 

5 

4 

676 

6012 

6050 

6057 

6065 

6072 

6080 

6087 

6095 

6103 

6110 

6 

4 

577 

6118 

6125 

6133 

6140 

614S 

6155 

6163 

6170 

6178 

6185 

7 

5 

578 

6193 

6200 

6208 

6215 

6223 

6230 

6238 

6245 

6253 

6260 

8 

6 

579 

6268 

6275 

6283 

6290 

6298 

6305 

6313 

6320 

6328 

6335 

9 

6 

No. 

0 

1 

2 

3 

4 1 

5 

6 

7 

8 

9 





















































156 


Logarithms of Numbers. 


No. 5800 to 6400. Logarithms. Log. 76343 to 80618. 


No. 

0 

1 1 

2 

3 

4 

5 

6 

7 

8 

9 


8 

580 

|76343 

76350 

76358 

76365 

76373 

76380 

76388 

76395 

76403 

76410 

l 

1 

581 

6418 

6425 

6433 

6440 

6448 

6455 

6462 

6470 

6477 

6485 

2 

2 

582 

6492 

6500 

6507 

.6515 

6522 

6530 

6537 

6545 

6552 

6559 

3 

2 

583 

6567 

6574 

6582 

6589 

6597 

6604 

6612 

6619 

6626 

6634 

4 

3 

581 

6641 

6649 

6656 

6664 

6671 

6678 

6686 

6693 

6701 

6708 

5 

4 

585 

76716 

76723 

76730 

76738 

76745 

76753 

76760 

76768 

76775 

76782 

6 

5 

586 

6790 

6797 

6S05 

6812 

6819 

6827 

6834 

6842 

6849 

6856 

7 

6 

587 

6864 

6871 

6879 

6886 

6893 

6901 

6908 

6916 

6923 

6930 

8 

6 

588 

6938 

6945 

6953 

6960 

6967 

6975 

6982 

6989 

6997 

7004 

9 

7 

589 

7012 

7019 

7026 

7034 

7041 

7048 

7056 

7063 

7070 

7078 



590 

77085 

77093 

77100 

77107 

77115 

77122 

77129 

77137 

77144 

77151 



591 

7159 

7166 

7173 

7181 

7188 

7195 

7203 

7210 

7217 

7225 



592 

7232 

7240 

7247 

7254 

7262 

7269 

7276 

7283 

7291 

7298 



593 

7305 

7313 

7320 

7327 

7335 

7342 

7349 

.7357 

7364 

7371 



594 

7379 

7386 

7393 

7401 

7408 

7415 

7422 

7430 

7437 

7444 



595 

77452 

77459 

77466 

77474 

77481 

77488 

77495 

77508 

77510 

77517 



596 

7525 

7532 

7539 

7546 

7554 

7561 

7568 

7576 

7583 

7590 



597 

7597 

7605 

7612 

7619 

7627 

7634 

7641 

7648 

7656 

7663 



598 

7670 

7677 

7685 

7692 

7699 

7706 

7714 

7721 

7728 

7735 



599 

7743 

7750 

7767 

7764 

7772 

7779 

7786 

7793 

7801 

7808 



600 

77815 

77822 

77830 

77837 

77844 

77851 

77859 

77866 

77873 

778S0 



601 

7887 

7895 

7902 

7909 

7916 

7924 

7931 

7938 

7945 

7952 



602 

7960 

7967 

7974 

7981 

7988 

7996 

8003 

8010 

8017 

8025 



603 

8032 

8039 

8046 

8053 

8061 

8068 

8075 

8082 

8089 

8097 



604 

8104 

8111 

8118 

8125 

8132 

8140 

8147 

8154 

8161 

S168 


V 

605 

78176 

78183 

78190 

78197 

78204 

78211 

78219 

78226 

78233 

78240 

1 

4 

1 

606 

8247 

8254 

8262 

8269 

8276 

8283 

8290 

8297 

8305 

8312 

9 


607 

8319 

8326 

8333 

8340 

8347 

8355 

8362 

8369 

8376 

8383 

9 

o 

608 

8390 

8398 

8405 

8412 

8419 

8426 

8433 

8440 

8447 

8455 

a 

Q 

609 

8462 

8469 

8476 

8483 

8490 

8497 

8504 

8512 

8519 

8526 


A 

610 

78533 

78540 

78547 

78554 

78561 

78569 

78576 

78583 

78590 

78597 


A 

611 

8604 

8611 

8618 

8625 

8633 

8640 

8647 

8654 

8661 

8668 

7 


612 

8675 

8682 

8689 

8696 

8704 

8711 

8718 

8725 

8732 

8739 

0 

0 

613 

8746 

8753 

8760 

8767 

8774 

8781 

8789 

8796 

8803 

8810 

o 


614 

8817 

8824 

8831 

8838 

8845 

8852 

8859 

8866 

8873 

8880 



615 

78888 

78895 

78902 

78909 

78916 

78023 

78930 

78937 

78944 

78951 



616 

8958 

8965 

8972 

8979 

8986 

8993 

9000 

9007 

9)14 

9021 



617 

9029 

9036 

9043 

9050 

9057 

9064 

9071 

9078 

9085 

9092 



618 

9099 

9106 

9113 

9120 

9127 

9134 

9141 

9148 

9155 

9162 



619 

9169 

9176 

9183 

9190 

9197 

9204 

9211 

9218 

9225 

9232 



620 

79239 

79246 

79253 

79260 

79267 

79274 

79281 

79288 

79295 

79302 



621 

9309 

9316 

9323 

9330 

9337 

9344 

9351 

935.8 

9365 

9372 



622 

9379 

9386 

9393 

9400 

9407 

9414 

9421 

9428 

9435 

9442 



623 

9449 

9456 

9463 

9470 

9477 

9484 

9491 

9498 

9505 

9511 



624 

9518 

9525 

9532 

9539 

9546 

9553 

9560 

9567 

9574 

9581 



a 25 

79588 

79595 

79602 

79609 

79616 

79623 

79630 

79637 

79644 

79650 



([26 

9657 

9664 

9671 

9678 

9685 

9692 

9699 

9706 

9713 

9720 



627 

9727 

9734 

9741 

9748 

9754 

9761 

9768 

9775 

9782 

9789 



628 

9796 

9803 

9810 

9817 

9824 

9831 

9837 

9844 

9851 

9858 



629 

9865 

9872 

9879 

9886 

9893 

9900 

9906 

9913 

9920 

9927 



630 

79934 

79911 

7994S 

79955 

79962 

79969 

79975 

79982 

79989 

79996 


6 

631 

80003 

80010 

80017 

80024 

80030 

80037 

80044 

80051 

80058 

80065 

1 

1 

632 

0072 

0079 

0085 

0092 

0099 

0106 

0113 

0120 

0127 

0134 

2 

1 

633 

0140 

0147 

0154 

0161 

0168 

0175 

0182 

0188 

0195 

0202 

3 

2 

634 

0209 

0216 

0223 

0229 

0236 

0243 

0250 

0257 

0264 

0271 

4 

2 

635 

80277 

80284 

8! 1291 

80298 

80305 

80312 

80318 

80325 

80332 

80339 

5 

3 

636 ; 

0346 

0353 

0359 

0366 

0373 

0380 

0387 

0393 

0400 

0407 

6 

4 

637; 

0414 

0421 

0428 

0434 

0441 

0448 

0455 

0462 

0468 

0475 

7 

4 

638 

0482 

0489 

0496 

0502 

0509 

0516 

0523 

0530 

0536 

0543 

8 

5 

639 

0550 

0557 

0564 

0570 

0577 

0584 

0591 

0598 

0604 

0611 

9 

5 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 







































Logarithms of Numbers. 


157 


No. 6400 to 7000. 


Logarithms. I-iOg. 80618 to 84510. 


No. 1 

0 

1 

2 | 

3 I 

4 

5 

6 

7 

8 

9 



610 i 

80618 

80625 

80632 

80638 

80645 

80652 

80659 

80665 

80672 

80679 


7 

641 > 

0686 

0693 

0699 

0706 

0713 

0720 

0726 

0733 

0740 

0747 

l 

1 

642 

0754 

0760 

0767 

0774 

0781 

0787 

0791 

0801 

0808 

0814 

2 

1 

G43 

0821 

0828 

0835 

0841 

0S48 

0855 

0862 

0868 

0875 

0882 

3 

2 

G44 

0889 

0895 

0902 

0909 

0916 

0922 

0929 

0936 

0943 

0949 


3 

645 i 

80956 

80963 

80969 

80976 

80983 

80990 

80996 

81003 

81010 

81017 

5 

4 

646 

1023 

1030 

1037 

1043 

1050 

1057 

1064 

1070 

1077 

1081 

6 

4 

647 

1090 

1097 

1104 

mi 

1117 

1124 

1131 

1137 

1144 

1151 

7 

5 

648 

1158 

1164 

1171 

1178 

1184 

1191 

1198 

1204 

1211 

1218 

8 

6 

649 

1224 

1231 

1238 

1245 

1251 

1258 

1265 

1271 

1278 

1285 

9 

6 

650 

81291 

81298 

81305 

81311 

S1318 

81325 

81331 

81338 

81345 

81351 



651 

1358 

1365 

1371 

7378 

1385 

1391 

1398 

1405 

1411 

1418 



652 

1425 

1431 

1438 

1445 

1451 

1458 

1465 

1471 

1478 

1485 



653 

1491 

1498 

1505 

1511 

1518 

1525 

1531 

3538 

1544 

1551 



654 

1558 

1564 

1571 

1578 

1584 

1591 

1598 

1604 

1611 

1617 



655 

81624 

81631 

81637 

81644 

81651 

81657 

81664 

81671 

81677 

81684 



656 

1690 

1697 

1704 

1770 

1717 

1723 

1730 

1737 

1743 

1750 



657 

1757 

1763 

1770 

1776 

1783 

1790 

1796 

1803 

1809 

1816 



658 

1823 

1829 

1836 

1842 

1819 

1856 

1862 

1869 

1875 

1882 



659 

1889 

1895 

1902 

1908 

1915 

1921 

1928 

1935 

1941 

1948 



660 

81954 

81961 

81968 

81974 

81981 

81987 

81994 

82000 

82007 

82014 



661 

2020 

2027 

2033 

2040 

2046 

2053 

2060 

*2066 

2073 

2079 



662 

2086 

2092 

2099 

2105 

2112 

2119 

2125 

2132 

2138 

2145 



663 

2151 

2158 

2164 

2171 

217S 

2184 

2191 

2197 

2204 

2216 



664 

2217 

2223 

2230 

2236 

2243 

2249 

2256 

2263 

2269 

2276 



665 

82282 

82289 

82295 

82302 

82308 

82315 

82321 

82328 

82334 

82341 



666 

2347 

2354 

2360 

2367 

2373 

2380 

2387 

2393 

2400 

2406 



667 

2413 

2419 

2426 

2432 

2439 

2445 

2452 

2458 

2463 

2471 



668 

2478 

2484 

2491 

2497 

2504 

2510 

2517 

2523 

2530 

2536 



669 

2543 

2549 

2556 

2562 

2569 

2575 

2582 

2588 

2595 

2601 



670 

82607 

82614 

82620 

82627 

82633 

82640 

82616 

82653 

82659 

82666 



671 

2672 

2679 

2685 

2692 

2698 

2705. 

2711 

2718 

2724 

2730 



672 

2737 

2743 

2750 

2756 

2763 

2769 

2776 

2782 

2789 

2795 



673 

2802 

2808 

2814 

2821 

2827 

2834 

2840 

2847 

2853 

2860 



674 

2866 

2872 

2879 

2885 

2892 

2898 

2905 

2911 

2918 

2921 



675 

82930 

82937 

82943 

82950 

82956 

82963 

82969 

82975 

82982 

82988 



676 

2995 

3001 

3008 

3014 

3020 

3027 

3033 

3040 

3046 

3052 



677 

3059 

3065 

3072 

3078 

3085 

3091 

3097 

3104 

3110 

3117 



678 

3123 

3129 

3136 

3142 

3149 

3155 

3161 

3168 

3174 

3181 



679 

3187 

3193 

3200 

3206 

3213 

3219 

3225 

3232 

3238 

3245 



680 

83251 

83257 

83264 

83270 

83276 

83283 

83289 

83296 

83302 

83308 



681 

3315 

3321 

3327 

3334 

3340 

3347 

3353 

3359 

3366 

3372 



682 

3378 

3385 

3391 

3398 

3404 

3410 

3117 

3423 

3429 

3436 



683 

3442 

3448 

3455 

3461 

3467 

3474 

3480 

3487 

3493 

3499 



684 

3506 

3512 

3518 

3525 

3531 

3537 

3544 

3550 

3556 

3563 



685 

83569 

83575 

83582 

83588 

83594 

83601 

83607 

83613 

83620 

83626 


6 

686 

3632 

3639 

3645 

3651 

3658 

3664 

3670 

3677 

3683 

3689 

1 

1 

687 

3696 

3702 

3708 

3715 

3721 

3727 

3734 

3740 

3746 

3753 

2 

1 

688 

3759 

3765 

3771 

3778 

3784 

3790 

3797 

3803 

3809 

3816 

3 

2 

689 

3822 

3828 

3835 

3841 

3847 

3853 

3860 

3866 

3872 

3879 

4 

2 

690 

83885 

83891 

83897 

83904 

83910 

83916 

83923 

83929 

83935 

83942 

5 

3 

691 

3948 

3954 

3960 

3967 

3973 

3979 

3985 

3992 

3998 

4004 

6 

4 

692 

4011 

4017 

4023 

4029 

4036 

4042 

4048 

4055 

4061 

4067 

7 

4 

693 

4073 

4080 

4086 

4092 

4098 

4105 

4111 

4117 

4123 

4130 

8 

5 

694 

4136 

4142 

4148 

4155 

4161 

4167 

4173 

4180 

4186 

4192 

9 

5 

695 

84198 

84205 

84211 

84217 

84223 

84230 

84236 

84242 

84248 

84255 



696 

4261 

4267 

4273 

4280 

4286 

4292 

429S 

4305 

4311 

4317 



697 

4323 

4330 

4336 

4342 

4348 

4354 

4361 

4387 

4373 

4379 



698 

4386 

4392 

4398 

4404 

4410 

4417 

4423 

4429 

4435 

4442 



699 

4448 

4454 

4460 

4466 

4473 

4479 

4485 

4491 

4497 

4504 



No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


































Logarithms or Numbers 


'158 


So. 7000 to 7600. 


Logarithms. 

Log. 84510 to 88081 

No. 

0 

1 

2 

3 

1 4 

5 

I 6 

j t-i 

i 

8 

9 


700 

84510 

S4516 

84522 

84528 

81535 

84541 

84547 

! 84553 

84559 

84566 


701 

j 4572 

4578 

4584 

4590 

4597 

4603 

4609 

4615 

4621 

4628 

1 

702 

4634 

4640 

4646 

4652 

465S 

4665 

4671 

4677 

46S3 

4689 

2 

703 

4696 

4702 

4708 

4714 

4720 

4726 

4733 

4739 

4745 

4751 

3 

704 

4757 

4763 

4770 

4776 

4782 

4788 

4794 

4800 

4807 

4813 

4 

705 

84819 

84825 

84831 

84837 

84844 

84850 

84856 

84862 

84868 

84874 

5 

706 

4880 

4887 

4893 

4899 

4905 

4911 

4917 

4924 

4930 

4936 

6 

707 

4942 

4948 

4954 

4960 

4967 

4973 

4979 

4985 

4991 

4997 

7 

70S 

5003 

5009 

5010 

5022 

5028 

5034 

5040 

5046 

5052 

5058 

8 

709 

5065 

5071 

5077 

50S3 

5089 

5095 

5101 

5107 

5114 

5120 

9 

710 

85126 

85132 

85138 

85144 

85150 

85156 

85163 

85169 

85175 

85181 


711 

5187 

5193 

5199 

5205 

5211 

5217 

5224 

5230 

5236 

5242 


712 

5248 

5254 

5260 

5266 

5272 

5278 

5285 

5291 

5297 

5303 


713 

5309 

5315 

5321 

5327 

.5333 

5339 

5345 

5352 

5358 

5364 


714 

5370 

5376 

5382 

5388 

5394 

5400 

5406 

5412 

5418 

5425 


715 

85431 

85437 

85443 

85449 

85455 

85461 

85437 

85473 

85479 

85485 


716 

5491 

5497 

5503 

5509 

5516 

5522 

5528 

5534 

5510 

5546 


717 

5552 

5558 

5564 

5570 

5576 

5582 

5588 

5594 

5600 

5606 


718 

5612 

5618 

5625 

5631 

5637 

5643 

5649 

5655 

5661 

5667 


719 

5673 

5679 

5685 

5691 

5697 

5703 

5709 

5715 

5721 

5727 


720 

85733 

85739 

85745 

85751 

85757 

85763 

85769 

85775 

85781 

85788 


721 

5794 

5800 

5806 

5812 

5818 

5824 

5830 

5836 

5842 

5848 


722 

5854 

5860 

5866 

5872 

5878 

5884 

5890 

5896 

5902 

5908 


723 

5914 

5920 

5926 

5932 

5938 

5944 

5950 

5956 

5962 

5968 


724 

5974 

5980 

5986 

5992 

5998 

6004 

6010 

6016 

6022 

6028 


725 

86034 

86040 

86046 

86052 

86058 

86064 

86070 

86076 

86082 

86088 

1 

726 

6094 

6100 

6106 

6112 

6118 

6124 

6130 

6136 

6141 

6147 

<? 

727 

6153 

6159 

6165 

6171 

6177 

6183 

6189 

6195 

6201 

6207 

3 

728 

6213 

6219 

6225 

6231 

6237 

6243 

6249 

6255 

6261 

6267 

A 

729 

6273 

6279 

6285 

6291 

6297 

6303 

6308 

6314 

6320 

6326 

5 

A 

730 

86332 

86338 

86344 

86350 

86356 

86362 

86368 

86374 

86380 

86386 

731 

6392 

6398 

6404 

6410 

6415 

6421 

6427 

6433 

6439 

6445 

7 

732 

6451 

6457 

6463 

6469 

6475 

6481 

6487 

6493 

6499 

6504 

8 

9 

733 

6510 

6516 

6522 

6528 

6534 

6540 

6546 

6552 

6558 

6564 

734 

6570 

6576 

6581 

6587 

6593 

6599 

6605 

6611 

6617 

6623 

735 

86629 

86635 

86641 

86646 

86652 

86658 

86664 

86670 

86676 

86682 


736 

6688 

6694 

6700 

6705 

6711 

6717 

6723 

6729 

6735 

6741 


737 

6747 

6753 

6769 

6764 

6770 

6776 

6782 

6788 

6794 

6800 


738 

6806 

6812 

6817 

6823 

6829 

6835 

6841 

6847 

6853 

6859 


730 

6864 

6S70 

6876 

6882 

6888 

6894 

6900 

6906 

6911 

6917 


740 

86923 

86929 

86935 

86941 

86947 

86953 

86958 

86964 

S6970 

86976 


741 

6982 

6988 

6994 

6999 

7005 

7011 

7017 

7023 

7029 

7035 


742 

7040 

7046 

7052 

7058 

7064 

7070 

7075 

7081 

7087 

7093 


743 

7099 

7105 

7111 

7116 

7122 

7128 

7134 

7140 

7146 

7151 


744 

7157 

7163 

7169 

7175 

7181 

7186 

7192 

7198 

7204 

7210 


745 

87216 

87221 

87227 

87233 

87239 

87245 

87251 

87256 

87262 

87268 


746 

7274 

7280 

7286 

7291 

7297 

7303 

7309 

7315 

7320 

7326 


747 

7332 

7338 

7344 

7349 

7355 

7361 

7367 

7373 

7379 

7384 


748 

7390 

7396 

7402 

7408 

7413 

7419 

7425 

7431 

7437 

7442 


749 

7448 

7454 

7460 

7466 

7471 

7477 

7483 

7489 

7495 

7500 


750 

87506 

87512 

87518 

87523 

87529 

87535 

87541 

87547 

87552 

S7558 


751 

7564 

7570 

7576 

7581 

7587 

7593 

7599 

7604 

7610 

7616 

1 

752 

7622 

7628 

7633 

7639 

7645 

7651 

7656 

7662 

7668 

7674 

2 

753 

7679 

7685 

7691 

7697 

7703 

7708 

7714 

7720 

7726 

7731 

3 

754 

7737 

7743 

7749 

7754 

7760 

7766 

7772 

7777 

7783 

7789 

4 

755 

87795 

87800 

87806 

87812 

87818 

87823 

87829 

87835 

87841 

87848 

5 

756 

7852 

7858 

7864 

7869 

7875 

7881 

7887 

7892 

7898 

7904 

6 

757 

7910 

7915 

7921 

7927 

7933 

7938 

7944 

7950 

7955 

7961 

7 

758 

7967 

7973 

7978 

7984 

7990 

7996 

8001 

8007 

8013 

8018 

8 

759 

8024 

8030 

8036 

8041 

8047 

8053 

8058 

8064 

8070 

8076 

9 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



7 

1 

1 

2 

3 

4 
4 


6 

1 

1 

2 

2 

3 

4 

4 

5 
5 


5 

1 

1 

2 

2 

3 

3 

4 

4 

5 


o o* c* 


















































Logarithms or Numbers, 


159 


No. /GOO to 8200. Logarithms. Log. 88081 to 91381. 



No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


ft 


700 

88081 

88087 

88093 

88098 

88104 

88110 

88116 

88121 

88127 

88133 

\ 



761 

8138 

8144 

8150 

8156 

8161 

8167 

8173 

8178 

8184 

8190 

2 

x 


762 

8195 

8201 

8207 

8213 

8218 

8224 

8230 

8235 

8241 

8247 


2 


763 

8252 

8258 

8264 

8270 

8275 

8281 

8287 

8292 

8298 

8304 

4 

2 


764 

8309 

8315 

8321 

8326 

8332 

8338 

8343 

8349 

8355 

8360 


3 


765 

88366 

88372 

88377 

88383 

88389 

88395 

88400 

88406 

88412 

88417 


4 


7 66 

8123 

8429 

8434 

440 

8446 

8451 

8457 

8463 

8468 

8474 


4 


767 

8480 

8485 

8491 

8497 

8502 

8508 

8513 

8519 

8525 

8530 

8 

5 


768 

8536 

8542 

8547 

8553 

8559 

8564 

8570 

8576 

8581 

8587 

9 

5 


769 

8593 

8598 

8604 

8610 

8615 

8621 

8627 

8632 

8638 

8643 




770 

88649 

88655 

88660 

88666 

88672 

88677 

88683 

88689 

88694 

88700 




771 

8705 

8711 

8717 

8722 

8728 

8734 

8739 

8745 

8750 

8756 




772 

8762 

8767 

8773 

8779 

8784 

8790 

8795 

8801 

8807 

8812 




773 

8818 

8824 

8829 

8835 

8840 

8846 

8852 

8857 

8863 

8868 




774 

8874 

8880 

8885 

8891 

8897 

8902 

8908 

8913 

8919 

8925 




775 

8" 930 

88936 

88941 

88947 

88953 

88958 

88964 

88969 

88975 

88981 




776 

8986 

8992 

8997 

9003 

9009 

9014 

9020 

9025 

9031 

9037 




777 

9042 

9048 

9053 

9059 

9064 

9070 

9076 

9081 

9087 

9092 




778 

9098 

9104 

9109 

9115 

9120 

9126 

9131 

9137 

9143 

9148 




779 

9154 

9159 

9165 

9170 

9176 

9182 

9187 

9193 

9198 

9204 




780 

89209 

89215 

89221 

89226 

89232 

89237 

89243 

89248 

89254 

89260 




781 

9205 

9271 

9276 

9282 

9287 

9293 

9298 

9304 

9310 

9315 




782 

9321 

9326 

9332 

9337 

9343 

9348 

9354 

9360 

9365 

9371 




783 

9376 

9382 

9387 

9393 

9398 

9404 

9409 

9415 

9421 

9426 




7 84 

9432 

9437 

9443 

9448 

9454 

9459 

9465 

9470 

9476 

9481 




785 

89487 

89492 

89498 

89504 

89509 

£9515 

S9520 

89526 

89531 

89537 




786 

9542 

9548 

9553 

9559 

9564 

9570 

9575 

9581 

9586 

9592 




787 

9597 

9603 

9609 

9614 

9620 

9625 

9631 

9636 

9642 

9647 




788 

9653 

9658 

9664 

9669 

9675 

9680 

9686 

9691 

9697 

9702 




789 

9708 

9713 

9719 

9724 

9730 

9735 

9741 

9746 

9752 

9757 




790 

89763 

89768 

89774 

89779 

89785 

89790 

89796 

89801 

S9S07 

89812 




791 

9818 

9823 

9829 

9834 

9840 

9845 

9851 

9856 

9862 

9867 




792 

9873 

9878 

9883 

9889 

9894 

9900 

9905 

9911 

9916 

9922 




793 

9927 

9933 

9938 

9944 

9949 

9955 

9960 

9966 

9971 

9977 




794 

9982 

9988 

9993 

9998 

90004 

90009 

90015 

90020 

90026 

90031 




795 

90037 

90042 

90048 

90053 

90059 

90064 

90069 

90075 

90080 

90086 




796 

0091 

0097 

0102 

0108 

0113 

0119 

0124 

0129 

0135 

0140 




797 

0146 

0 51 

0157 

0162 

0168 

0173 

0179 

0184 

0189 

0195 




798 

0200 

0206 

0211 

0217 

0222 

0227 

0233 

0238 

0244 

0249 




799 

0255 

0260 

0266 

0271 

0276 

0282 

0287 

0293 

0298 

0304 




800 

90309 

90314 

90320 

90325 

90331 

90336 

90342 

90347 

90352 

90358 




801 

0363 

0369 

0374 

0380 

03S5 

0390 

0396 

0401 

0407 

0112 




802 

0417 

0423 

0428 

0434 

0439 

0445 

0450 

0455 

0461 

0466 




803 

0172 

0477 

0482 

0488 

0493 

0499 

0504 

0509 

0515 

0520 




804 

0526 

0531 

0536 

0542 

0547 

0553 

0558 

0563 

0569 

0574 




805 

90580 

90585 

90590 

90596 

90601 

90607 

90612 

90617 

90623 

90628 


5 


806 

0634 

0639 

0644 

0650 

0655 

0660 

0666 

0671 

0677 

0682 

i 

1 


807 

0687 

0093 

0698 

0703 

0709 

0714 

0720 

0725 

0730 

0736 

2 

1 


8'>8 

0741 

0747 

0752 

0757 

0763 

0768 

0773 

0779 

0784 

0789 

3 

2 


809 

0795 

0800 

0806 

0811 

0816 

0822 

0827 

0832 

0838 

0843 

4 

2 


810 

90849 

90854 

90859 

90865 

90870 

90875 

90881 

90886 

90891 

90897 

5 

3 


811 

0902 

0907 

0913 

0918 

0924 

0929 

0934 

0940 

0945 

0950 

6 

3 


812 

0956 

0961 

0966 

0972 

0977 

0982 

0988 

0993 

0998 

1004 

7 

4 


813 

1009 

1014 

1020 

1025 

1030 

1036 

1041 

1046 

1052 

1057 

8 

4 


814 

1062 

1068 

1073 

1078 

1084 

1089 

1094 

1100 

1105 

1110 

9 

5 


815 

91116 

91121 

91126 

91132 

91137 

91142 

91148 

91153 

91158 

91164 




816 

1169 

1174 

1180 

1185 

1190 

1196 

1201 

1206 

1212 

1217 




817 

1222 

1228 

1233 

1238 

1243 

1249 

1254 

1259 

1265 

1270 




818 

1275 

1281 

1286 

1291 

1297 

1302 

1307 

1312 

1318 

1323 




819 

1328 

1331 

1339 

1344 

1350 

1355 

1360 

1365 

1371 

1376 




No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 





























160 Logarithms of Numbers. 


No. 8200 to 8800. 


Logarithms. L/Og. 91381 to 94448. 


No. 

0 

1 

2 

3 i 

4 

5 

6 

7 

8 

9 


6 

820 s 

91381 

91387 

91392 

91397 

91403 

91408 

91413 

91418 

91424 

91429 

1 

1 

821 

1434 

1440 

1445 

1450 

1455 

1461 

1466 

1471 

1477 

1482 

2 

1 

822 

1487 

1492 

1498 

1503 

1508 

1514 

1519 

1524 

1529 

1535 

3 

2 

823 ! 

1540 

1545 

1551 

1556 

1561 

1566 

1572 

1577 

1582 

1587 

4 

2 

824 

1593 

1598 

1603 

1609 

1614 

1619 

1624 

1630 

1635 

1640 

5 

3 

825 

91645 

91651 

91656 

91661 

91666 

91672 

91677 

91682 

91687 

91693 

6 

4 

826 

1698 

1703 

1709 

1714 

1719 

1724 

1730 

1735 

1740 

1745 

7 

4 

827 

1751 

1756 

1761 

1766 

1772 

1777 

1782 

1787 

1793 

1798 

8 

5 

828 

1803 

1808 

1814 

1819 

1824 

1829 

1834 

1840 

1845 

1850 

9 

5 

829 

1855 

1861 

1866 

1871 

1876 

1882 

1887 

1892 

1897 

1903 



83C 

9190S 

91913 

91918 

91924 

91929 

91934 

91939 

91944 

91950 

91955 



831 

1960 

1965 

1971 

1976 

1981 

1986 

1991 

1997 

2002 

2007 



832 

2012 

2018 

2023 

2028 

2033 

2038 

2044 

2049 

2054 

2059 



833 

2065 

2070 

2075 

2080 

2085 

2091 

2096 

2101 

2106 

2111 



834 

2117 

2122 

2127 

2132 

2137 

2143 

2148 

2153 

2158 

2163 



835 

92169 

92174 

92179 

92184 

92189 

92195 

92200 

92205 

92210 

92215 



836 

2221 

2226 

2231 

2236 

2241 

2247 

2252 

2257 

2262 

2267 



837 

2273 

2278 

2283 

2288 

2293 

2298 

2304 

2309 

2314 

2319 



838 

2324 

2330 

2335 

2340 

2345 

2350 

2355 

2361 

2366 

2371 



839 

2376 

2381 

2387 

2392 

2397 

2402 

2407 

2412 

2418 

2423 



840 

92428 

92433 

92438 

92443 

92449 

92454 

92459 

92464 

92469 

92474 



841 

2480 

2485 

2490 

2495 

2500 

2505 

2511 

2516 

2521 

2526 



842 

2531 

2536 

2542 

2547 

2552 

2557 

2562 

2567 

2572 

2578 



843 

2583 

2588 

2593 

2598 

2603 

2609 

2614 

2619 

2624 

2629 



844 

2634 

2639 

2645 

2650 

2655 

2660 

2665 

2670 

2675 

2681 


5 

845 

92686 

92691 

92696 

92701 

92706 

92711 

92716 

92722 

92727 

92732 

1 

1 

846 

2737 

2742 

2747 

2752 

2758 

2763 

2768 

2773 

2778 

2783 

2 

1 

847 

2788 

2793 

2799 

2804 

2809 

2814 

2819 

2824 

2829 

2S34 

3 

2 

848 

2840 

2845 

2850 

2855 

2860 

2865 

2870 

2875 

2881 

2886 

4 

2 

849 

2891 

2896 

2901 

2906 

2911 

2916 

2921 

2927 

2932 

2937 

5 

Q 

85.) 

92942 

92947 

92952 

92957 

92962 

92967 

92973 

92978 

92983 

92988 

6 

3 

851 

2993 

2998 

3003 

3008 

3013 

3018 

3024 

3029 

3034 

3039 

7 

4 

852 

3044 

3049 

3054 

3059 

3064 

3069 

3075 

3080 

3085 

3090 

8 

4 

853 

3095 

3100 

3105 

3110 

3115 

3120 

3125 

3131 

3136 

3141 

9 

5 

854 

3146 

3151 

3156 

3161 

3166 

3171 

3176 

3181 

3186 

3192 



855 

93197 

93202 

93207 

93212 

93217 

93222 

93227 

93232 

93237 

93242 



856 

3247 

3252 

3258 

3263 

3268 

3273 

3278 

3283 

3288 

3293 



857 

3298 

3303 

3308 

3313 

3318 

3323 

3328 

3334 

3339 

3344 



858 

3349 

3354 

3359 

3364 

3369 

3374 

3379 

3384 

3389 

3394 



859 

3399 

3404 

3409 

3414 

3420 

3425 

3430 

3435 

3440 

3445 



860 

93450 

93455 

93460 

93465 

93470 

93475 

93480 

93485 

93490 

93495 



861 

3500 

3505 

3510 

3515 

3520 

3526 

3531 

3536 

3541 

3546 



862 

3551 

3556 

3561 

3566 

3571 

3576 

3581 

3586 

3591 

3596 



863 

3601 

3606 

3611 

3616 

3621 

3626 

3631 

3636 

3641 

3646 



864 

3651 

3656 

3661 

3666 

3671 

3676 

3682 

3687 

3692 

3697 



865 

93702 

93707 

93712 

93717 

93722 

93727 

93732 

93737 

93742 

93747 



866 

3752 

3757 

3762 

3767 

3772 

3777 

3782 

3787 

3792 

3797 



867 

3802 

3807 

3812 

3817 

3822 

3827 

3832 

3837 

3842 

3847 



868 

3852 

3857 

3862 

3867 

3872 

3877 

3882 

3887 

3892 

3897 



869 

3902 

3907 

3912 

3917 

3922 

3927 

3932 

3937 

3942 

3947 



870 

93952 

93957 

93962 

93967 

93972 

93977 

93982 

93987 

93992 

93997 


4: 

871 

4002 

4007 

4012 

4017 

4022 

4027 

4032 

4037 

4042 

4047 

1 

0 

872 

4052 

4057 

4062 

4067 

4072 

4077 

4082 

4086 

4091 

4096 

2 

1 

873 

4101 

4106 

4111 

4116 

4121 

4126 

4131 

4136 

4141 

4146 

3 

1 

874 

4151 

4156 

4161 

4166 

4171 

4176 

4181 

4186 

4191 

4196 

4 

2 

875 

94201 

94206 

94211 

94216 

94221 

94226 

94231 

94236 

94240 

94245 

5 

2 

876 

4250 

4255 

4260 

4265 

4270 

4275 

4280 

4285 

4290 

4295 

6 

2 

877 

4300 

4305 

4310 

4315 

4320 

4325 

4330 

43&5 

4340 

4345 

7 

3 

878 

4349 

4354 

4359 

4364 

4369 

4374 

4379 

4384 

4389 

4394 

8 

3 

879 

4399 

4404 

4409 

4414 

4419 

4424 

4429 

4433 

4438 

4443 

9 

4 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 








































Logarithms of Numbers. 161 


No. 8800 to 9400. 


Logarithms. 

Log. 94448 to 97313 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



880 

4 94448 

94153 

94458 

94463 

94468 

94473 

94478 

94483 

94488 

94493 

1 


881 

4498 

4503 

4507 

4512 

4517 

4522 

4527 

4532 

4537 

4542 

9 

1 

882 

4517 

4552 

4557 

4562 

4567 

4571 

4576 

4581 

4586 

4591 

3 

9 

SS3 

4596 

4601 

4606 

4611 

4616 

4621 

4626 

4630 

4635 

4640 

A 

9 

881 

4645 

4650 

4655 

4660 

4665 

4870 

4675 

4680 

4685 

4689 

* 

3 

885 

94694 

94699 

94704 

94709 

91714 

91719 

94724 

94729 

94734 

94738 

ft 

Q 

8S6 

4743 

4748 

4753 

4758 

4763 

4768 

4773 

4778 

47S3 

4787 

7 

4 

887 

4792 

4797 

4802 

4807 

4812 

4817 

4822 

4827 

4832 

4836 

8 

A 

888 

4841 

4816 

4851 

4856 

4861 

4866 

4871 

4876 

4880 

4885 

4 ) 


889 

4890 

4895 

4900 

4905 

4910 

4915 

4919 

4924 

4929 

4934 



890 

94939 

94944 

94949 

94954 

94959 

94963 

94968 

94973 

94978 

94983 



891 

4988 

4993 

4998 

5002 

5007 

5012 

6017 

5022 

5027 

5032 



892 

6036 

5041 

5016 

5051 

5056 

5061 

5066 

5071 

5075 

5080 



893 

5085 

6090 

5095 

5100 

5105 

5109 

5114 

5119 

5124 

5129 



894 

5134 

5139 

5143 

5148 

5153 

5158 

5163 

5168 

5173 

5177 



895 

95182 

95187 

95192 

95197 

95202 

95207 

95211 

95216 

95221 

95226 



896 

5231 

5236 

5210 

5245 

5250 

5255 

5260 

5265 

5270 

5274 



897 

5279 

5284 

5289 

5294 

5299 

5303 

5308 

5313 

5318 

5323 



898 

5328 

5332 

5337 

5342 

5347 

5352 

5357 

5361 

5366 

5371 



899 

5376 

5381 

5386 

5390 

5395 

5400 

5405 

5410 

5415 

5419 



900 

95424 

95429 

95434 

95439 

95444 

95448 

95453 

95458 

95463 

95408 



901 

5472 

5477 

5482 

5487 

5492 

5497 

5501 

5506 

5511 

5516 



902 

6521 

5525 

5530 

5535 

5540 

5545 

5550 

5554 

5559 

5564 



903 

5569 

5574 

5578 

5583 

5588 

5593 

5598 

5602 

5007 

5612 



904 

5617 

5622 

5626 

5631 

5636 

5641 

5646 

5650 

5655 

5660 



905 

95665 

95670 

95674 

95679 

95684 

95689 

95694 

95698 

95703 

95708 



906 

5713 

5718 

5722 

5727 

5732 

5737 

5742 

5746 

5751 

5756 



9u7 

5761 

5766 

5770 

5775 

5780 

5785 

5789 

5794 

5799 

5804 



908 

5809 

5813 

5818 

5823 

5828 

5832 

5837 

5842 

5847 

5852 



009 

5856 

5861 

5866 

5871 

5875 

5880 

5885 

5890 

5895 

5899 



910 

95904 

95909 

95914 

95918 

95923 

95928 

95933 

95938 

95942 

95947 



911 

5952 

6957 

5961 

5966 

5971 

5976 

5980 

5985 

5990 

5995 



912 

5999 

60o4 

6009 

6014 

6019 

6023 

6028 

6033 

6038 

6042 



913 

6047 

6052 

6057 

6061 

6066 

6071 

6076 

6080 

6085 

6090 



914 

6095 

6099 

6104 

6109 

6114 

6118 

6123 

6128 

6133 

6137 



915 

961.42 

96147 

96152 

96156 

96161 

96166 

96171 

96175 

96180 

96185 



916 

6190 

6194 

6199 

6201 

6209 

6213 

6218 

6223 

6227 

6232 



917 

6237 

6242 

6216 

6251 

6256 

6261 

6265 

6270 

6275 

6280 



918 

6284 

6289 

6294 

6298 

6303 

6308 

6313 

6317 

6322 

6327 



919 

6332 

6336 

6341 

6346 

6350 

6355 

6360 

6365 

6369 

6374 



920 

96379 

96384 

96388 

96393 

96398 

96402 

96407 

96412 

96417 

90421 



921 

6426 

6431 

6435 

6440 

6445 

6450 

6454 

6459 

6464 

6468 



922 

6473 

6478 

6483 

6487 

6492 

6497 

6501 

6506 

6511 

6515 



923 

6520 

6523 

6530 

6534 

6539 

6544 

6548 

6553 

6558 

6562 



924 

6567 

6572 

6577 

6581 

6586 

6591 

6595 

6600 

6605 

6609 



925 

96614 

96619 

96624 

96628 

96633 

96638 

96642 

96647 

96652 

96656 


4 

926 

6661 

6666 

6670 

6675 

6680 

6685 

6689 

6694 

6699 

6703 

i 

0 

927 

6708 

6713 

6717 

6722 

6727 

6731 

6736 

6741 

6745 

6750 

2 

l 

928 

6755 

6759 

6764 

6769 

6774 

6778 

6783 

6788 

6792 

6797 

3 

l 

929 

6802 

6806 

6811 

6816 

6820 

6825 

6830 

6834 

6839 

6844 

4 

2 

930 

96848 

96853 

96858 

96S62 

96867 

96872 

96876 

96881 

96886 

90890 

5 

2 

931 

6895 

6900 

6904 

6909 

6914 

6918 

6923 

6928 

6932 

6937 

6 

2 

932 

6942 

6946 

6951 

6956 

6960 

6965 

6970 

6974 

6979 

6984 

7 

3 

933 

6983 

6993 

6997 

7002 

7007 

7011 

7016 

7021 

7025 

7030 

8 

3 

934 

7035 

7039 

7044 

7049 

7053 

7058 

7063 

7067 

7072 

7077 

9 

4 

935 

97081 

97086 

97090 

97095 

97100 

97104 

97109 

97114 

97118 

97123 



936 

7128 

7132 

7137 

7142 

7146 

7151 

7155 

7160 

7165 

7169 



937 

7174 

7179 

7183 

7188 

7192 

7197 

7202 

7206 

7211 

7216 



938 

7220 

7225 

7230 

7231 

7239 

7243 

7248 

7253 

7257 

7262 



939 

7267 

7271 

7276 

7280 

7285 

7290 

7294 

7299 

7304 

7308 



No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 




11 


































162 Logarithms of Numbers. 


No. 9400 to 10000. 


Logarithms. Log. 97313 to 99996 


No.] 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


5 

940 

97313 

97317 

97322 

97327 

97331 

97336 

97340 

97345 

97350 

97354 

1 

1 

941 

7359 

7364 

7368 

7373 

7377 

7382 

7387 

7391 

7396 

7400 

2 

1 

842 

7405 

7410 

7414 

7419 

7424 

7428 

7433 

7437 

7442 

7447 

3 

2 

943 

7451 

7456 

7460 

7465 

7470 

7474 

7479 

7483 

7488 

7493 

4 

2 

944 

7497 

7502 

7506 

7511 

7516 

7520 

7525 

7529 

7534 

7539 

5 

3 

945 

97543 

97548 

97552 

97557 

97562 

97566 

97571 

97575 

97580 

97585 

6 

3 

946 

7589 

7594 

7598 

7603 

7607 

7612 

7617 

7621 

7626 

7630 

7 

4 

947 

7635 

7640 

7644 

7649 

7653 

7658 

7663 

7667 

7672 

7676 

8 

4 

948 

7681 

7685 

7690 

7695 

7699 

7704 

7708 

7713 

7717 

7722 

9 

5 

919 

7727 

7731 

7736 

7740 

7745 

7749 

7754 

7759 

7763 

7768 



950 

97772 

97777 

97782 

97786 

97791 

97795 

97800 

97804 

97809 

97813 



951 

7818 

7823 

7827 

7832 

7836 

7841 

7845 

7850 

7855 

7859 



952 

7864 

7868 

7873 

7877 

7882 

7886 

7891 

7896 

7900 

7905 



953 

7909 

7914 

7918 

7923 

7928 

7932 

7937 

7941 

7946 

7950 



954 

7955 

7959 

7964 

7968 

7973 

7978 

7982 

7987 

7991 

7996 



955 

98000 

98005 

98009 

98014 

98019 

98023 

98028 

98032 

98037 

98041 



956 

8046 

8050 

8055 

8059 

8064 

8068 

8073 

8078 

8082 

8087 



957 

8091 

8096 

8100 

8105 

8109 

8114 

8118 

8123 

8127 

8132 



958 

8137 

8141 

8146 

8150 

8155 

8159 

8164 

8168 

8173 

8177 



959 

8182 

8186 

8191 

8195 

8200 

8204 

8209 

8214 

8218 

8223 



960 

98227 

98232 

98236 

98241 

98245 

98250 

98254 

98259 

98263 

98268 



961 

8272 

8277 

8281 

8286 

8290 

8295 

8299 

8304 

8308 

8313 



962 

8318 

8322 

8327 

8331 

8336 

8340 

8345 

8349 

8354 

8358 



963 

8363 

8367 

8372 

8376 

8381 

8385 

8390 

8394 

8399 

8403 



964 

8408 

8412 

8417 

8421 

8426 

8430 

8435 

8439 

8444 

8448 



965 

98453 

98457 

98462 

98466 

98471 

98475 

98480 

98484 

98489 

98493 



966 

8498 

8502 

8507 

8511 

8516 

8520 

8525 

8529 

8534 

8538 



967 

8543 

8547 

8552 

8556 

8561 

8565 

8570 

8574 

8579 

8583 



968 

8588 

8592 

8597 

8601 

8605 

8610 

8614 

8619 

8623 

8628 



969 

8632 

8637 

8641 

8646 

8650 

8655 

8659 

8664 

8668 

8673 



970 

98677 

98682 

98686 

98691 

98695 

98700 

98704 

98709 

98713 

98717 



971 

8722 

8726 

8731 

8735 

8740 

8744 

8749 

8753 

8758 

8762 



972 

8767 

8771 

87^6 

8780 

8784 

8789 

8793 

8798 

8802 

8807 



973 

8811 

8816 

8820 

8825 

8829 

8834 

8838 

8843 

8847 

8851 



974 

8856 

8860 

8865 

8869 

8874 

8878 

8883 

8887 

8892 

8896 



976 

98900 

98905 

98909 

98914 

98918 

98923 

98927 

98932 

98936 

98941 



976 

8945 

8949 

8954 

8958 

8963 

8967 

8972 

8976 

8981 

.8986 



977 

8989 

8994 

8998 

9003 

9007 

9012 

9016 

9021 

9025 

9029 



978 

9034 

9038 

9043 

9047 

9052 

9056 

9061 

9065 

9069 

9074 



979 

9078 

9083 

9087 

9092 

9096 

9100 

9105 

9109 

9114 

9118 



980 

99123 

99127 

99131 

99136 

99140 

99145 

99149 

99154 

99158 

99162 



981 

9167 

9171 

9176 

9180 

9185 

9189 

9193 

9198 

9202 

9207 



982 

9211 

9216 

9220 

9224 

9229 

9233 

9238 

9242 

9247 

9251 



983 

9255 

9260 

9264 

9269 

9273 

9277 

9282 

9286 

9291 

9295 



984 

9300 

9304 

9308 

9313 

9317 

9322 

9326 

9330 

9335 

9339 



985 

99344 

99348 

99352 

99357 

99361 

99366 

99370 

99374 

99379 

99383 


•1 

986 

9388 

9392 

9396 

9401 

9405 

9410 

9414 

9419 

9423 

9427 

1 

0 

987 

9432 

9436 

9441 

9445 

9449 

9454 

9458 

9463 

9467 

9471 

2 

1 

988 

9476 

9480 

9484 

9489 

9493 

9498 

9502 

9506 

9511 

9515 

3 

1 

989 

9520 

9524 

9528 

9533 

9537 

9542 

9546 

9550 

9555 

9559 

4 

2 

990 

99564 

99568 

99572 

99577 

99581 

99585 

99590 

99594 

99599 

99603 

5 

a 

991 

9607 

9612 

9616 

9621 

9625 

9629 

9634 

9638 

9642 

9647 

6 

2 

992 

9651 

9656 

9660 

9664 

9669 

9673 

9677 

9682 

9686 

9691 

7 

3 

993 

9695 

9699 

9704 

9708 

9712 

9717 

9721 

9726 

9730 

9734 

8 

3 

994 

9739 

9743 

9747 

9752 

9756 

9760 

9765 

9769 

9774 

9778 

9 

4 

995 

99782 

99787 

99791 

99795 

99800 

99804 

99808 

99813 

99817 

99822 



996 

9826 

9S30 

9835 

9839 

9843 

9848 

9852 

9856 

9861 

9865 



997 

9870 

9874 

9878 

9883 

9887 

9891 

9896 

9900 

9904 

9909 



998 

9913 

9917 

9922 

9926 

9930 

9935 

9939 

9944 

994S 

9952 



999 

9957 

9961 

9965 

9970 

9974 

9978 

9983 

9987 

9991 

9996 



No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 
























Logarithms Trigonometric. 


1G3 


0 h 

0° 



Logarithms. 


179° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

00 

0 

Iuf.Neg. 

Infinite. 

Inf. Nee;. 

Infinite. 

10.00000 

10.00000 

60 

60 

4 

1 

6.46373 

13.53627 

6.46373 

13.53627 

00000 

00000 

59 

56 

8 

2 

76476 

23524 

76476 

23524 

00000 

00000 

58 

52 

12 

3 

94085 

05915 

94085 

05915 

00000 

00000 

57 

48 

16 

4 

7.06579 

12.93421 

7.06579 

12.93421 

00000 

00000 

56 

44 

20 

5 

7.16270 

12.83730 

7.16270 

12.83730 

10.00000 

10.00000 

55 

40 

24 

6 

24188 

75812 

24188 

75812 

00000 

00000 

54 

36 

28 

7 

30882 

69118 

30882 

69118 

00000 

00000 

53 

32 

32 

8 

36682 

63318 

36682 

63318 

00000 

00000 

52 

28 

36 

9 

41797 

58203 

41797 

58203 

00000 

00000 

51 

24 

40 

10 

7.46373 

12.53627 

7.46373 

12.53627 

10.00000 

10.00000 

50 

20 

44 

11 

50512 

49488 

50512 

49488 

00000 

00000 

49 

16 

43 

12 

54291 

45709 

54291 

45709 

00000 

00000 

48 

12 

62 

13 

57767 

42233 

57767 

42233 

00000 

00000 

47 

8 

56 

14 

60985 

39015 

60986 

39014 

00000 

00000 

46 

4 

1 

15 

7.63982 

12.36018 

7.63982 

12.36018 

10.00000 

10.00000 

45 

59 

4 

16 

66784 

33216 

66785 

33215 

00000 

00000 

44 

56 

8 

17 

69417 

30583 

69418 

30582 

00001 

9.99999 

43 

52 

12 

18 

71900 

28100 

71900 

28100 

00001 

99999 

42 

48 

16 

19 

74248 

25752 

74248 

25752 

00001 

99999 

41 

44 

20 

20 

7.76475 

12.23525 

7.76476 

12.23524 

10.00001 

9.99999 

40 

40 

24 

21 

78594 

21406 

78595 

21405 

00001 

99999 

39 

36 

28 

22 

80615 

19385 

80615 

19385 

00001 

99999 

38 

32 

32 

23 

82545 

17455 

82546 

17454 

00001 

99999 

37 

28 

36 

24 

84393 

15607 

84394 

15606 

00001 

99999 

36 

24 

40 

25 

7.86166 

12.13834 

7.86167 

12.13833 

10.00001 

9.99999 

35 

20 

44 

26 

87870 

12130 

87871 

12129 

00001 

99999 

34 

16 

48 

27 

89509 

10491 

89510 

10490 

00001 

99999 

33 

12 

52 

28 

91088 

08912 

91089 

08911 

00001 

99999 

32 

8 

56 

29 

92612 

07388 

92613 

07387 

00002 

99998 

31 

4 

3 

30 

7.04084 

12.05916 

7.94086 

12.05914 

10.00002 

9.99998 

30 

58 

4 

31 

95508 

04492 

95510 

04490 

00002 

99998 

29 

56 

8 

32 

96887 

03113 

968S9 

03111 

00002 

99998 

28 

52 

12 

33 

98223 

01777 

98225 

01775 

00002 

99998 

27 

48 

16 

34 

99520 

00480 

99522 

00478 

00002 

99998 

26 

44 

20 

35 

8.00779 

11.99221 

8.00781 

11.99219 

10.00002 

9.99998 

25 

40 

24 

36 

02002 

97998 

02004 

97996 

00002 

99998 

24 

36 

28 

37 

03192 

96808 

03194 

96806 

00003 

99997 

23 

32 

32 

38 

04350 

95650 

04353 

95647 

00003 

99997 

22 

28 

36 

39 

05478 

94522 

05481 

94519 

00003 

99997 

21 

24 

40 

40 

8.06578 

11.93422 

8.06581 

11.93419 

10.00003 

9.99997 

20 

20 

44 

41 

07650 

92350 

07653 

92347 

00003 

99997 

19 

16 

48 

42 

08696 

91304 

08700 

91300 

00003 

99997 

18 

12 

52 

43 

09718 

90282 

09722 

90278 

00003 

99997 

17 

8 

56 

44 

10717 

89283 

10720 

89280 

00004 

99996 

16 

4 

3 

45 

8.11693 

11.88307 

8.11696 

11.88304 

10.00004 

9.99996 

15 

57 

4 

46 

12647 

87353 

12651 

87349 

00004 

99996 

14 

56 

8 

47 

13581 

86419 

13585 

86415 

00004 

99996 

13 

52 

12 

48 

14495 

85505 

14500 

85500 

00004 

99996 

12 

48 

16 

49 

15391 

84609 

15395 

84605 

00004 

99996 

11 

44 

20 

50 

8.16268 

11.83732 

8.16273 

11.83727 

10.00005 

9.99995 

10 

40 

24 

51 

17128 

82872 

17133 

82867 

00005 

99995 

9 

36 

28 

52 

17971 

82029 

17976 

82024 

00005 

99995 

8 

32 

32 

53 

18798 

81202 

18804 

81196 

00005 

99995 

7 

28 

36 

54 

19610 

80390 

19616 

80384 

00005 

99995 

6 

24 

40 

55 

8.20407 

11.79593 

8.20413 

11.79587 

10.00006 

9.99994 

5 

20 

44 

56 

21189 

78811 

21195 

78805 

00006 

99994 

4 

16 

48 

57 

21958 

78042 

21964 

78036 

00006 

99994 

O 

O 

12 

52 

58 

22713 

77287 

22720 

77280 

00006 

99994 

2 

8 

56 

59 

23456 

76544 

23462 

76538 

00006 

99994 

1 

4 

4 

60 

24186 

75814 

24192 

75808 

00007 

99993 

o 

56 

M. S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M. S. 

6 h [90° 







89°l 

iLi 






















164 Logarithms Trigonometric. 


o h 

1° 



Logarithms. 


178 c 

5 jll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

4 

0 

8.24186 

11.75814 

8.24192 

11.75808 

10.00007 

9.99993 

60 

5G 

4 

1 

24903 

75097 

24910 

75090 

00007 

99993 

59 

56 

8 

2 

25609 

74391 

25616 

74384 

00007 

99993 

58 

52 

12 

3 

26304 

73696 

26312 

73688 

00007 

99993 

57 

48 

16 

4 

26988 

73012 

26996 

73004 

00008 

99992 

56 

44 

20 

5 

8.27661 

11.72339 

8.27669 

11.72331 

10.00008 

9.99992 

55 

40 

24 

6 

28324 

71676 

28332 

7166S 

00008 

99992 

54 

36 

28 

7 

28977 

71023 

28986 

71014 

00008 

99992 

53 

32 

32 

8 

29621 

70379 

29629 

70371 

00008 

99992 

52 

28 

36 

9 

30255 

69745 

30263 

69737 

00009 

99991 

51 

24 

40 

10 

8.30879 

11.69121 

8.30888 

11.69112 

10.00009 

9.99991 

50 

20 

44 

11 

31495 

68505 

31505 

68495 

00009 

99991 

49 

16 

48 

12 

32103 

67897 

32112 

67888 

00010 

99990 

48 

12 

52 

13 

32702 

67298 

32711 

67289 

00010 

99990 

47 

8 

56 

14 

33292 

66708 

33302 

66698 

00010 

99990 

46 

4 

5 

15 

8.33875 

11.66125 

8.33886 

11.66114 

10.00010 

9-.99990 

45 

55 

4 

16 

34450 

65550 

34461 

65539 

00011 

99989 

44 

56 

8 

17 

35018 

64982 

35029 

64971 

00011 

99989 

43 

52 

12 

18 

35578 

64422 

35590 

64410 

00011 

99989 

42 

48 

16 

19 

36131 

63869 

36143 

63857 

00011 

99989 

41 

4-4 

20 

20 

8.36678 

11.63322 

8.36689 

11.63311 

10.00012 

9.99988 

40 

40 

24 

21 

37217 

62783 

37229 

62771 

00012 

99988 

39 

36 

28 

22 

37750 

62250 

37762 

62238 

00012 

99988 

3S 

32 

32 

23 

38276 

61724 

38289 

61711 

00013 

99987 

37 

28 

36 

24 

38796 

61204 

38809 

61191 

00013 

99987 

36 

24 

40 

25 

8.39310 

11.60690 

8.39323 

11.60677 

10.00013 

9.99987 

35 

20 

44 

26 

39818 

60182 

39832 

60168 

00014 

99986 

34 

16 

48 

27 

40320 

59680 

40334 

59666 

00014 

99986 

33 

12 

52 

28 

40816 

59184 

40830 

59170 

00014 

99986 

32 

8 

56 

29 

41307 

58693 

41321 

58679 

00015 

99985 

31 

4 

G 

30 

8.41792 

11.58208 

8.41807 

11.58193 

10.00015 

9.99985 

30 

54 

4 

31 

42272 

57728 

42287 

57713 

00015 

99985 

29 

56 

8 

32 

42746 

57254 

42762 

57238 

00016 

99984 

28 

52 

12 

33 

43216 

56784 

43232 

56768 

00016 

99984 

27 

48 

16 

34 

43680 

56320 

43696 

56304 

00016 

99984 

26 

44 

20 

35 

8.44139 

11.55861 

8.44156 

11.55844 

10.00017 

9.99983 

25 

40 

24 

36 

44594 

55406 

44611 

55389 

00017 

99983 

24 

36 

28 

37 

45044 

54956 

45061 

54939 

00017 

99983 

23 

32 

32 

38 

45489 

54511 

45507 

54493 

00018 

99982 

22 

28 

36 

39 

45930 

54070 

45948 

54052 

00018 

99982 

21 

24 

40 

40 

8.46366 

11.53634 

8.46385 

11.53615 

10.00018 

9.99982 

20 

20 

44 

41 

46799 

53201 

46817 

53183 

00019 

99981 

19 

16 

48 

42 

47226 

52774 

47245 

52755 

00019 

99981 

18 

12 

52 

43 

47650 

52350 

47669 

52331 

00019 

99981 

17 

8 

56 

44 

48069 

51931 

48089 

51911 

00020 

99980 

16 

4 

7 

45 

8.48485 

11.51515 

8.48505 

11.51495 

10.00020 

9.99980 

15 

53 

4 

46 

48896 

51104 

48917 

51083 

00021 

99979 

14 

56 

8 

47 

49304 

50696 

49325 

50675 

00021 

99979 

13 

52 

12 

48 

49708 

50292 

49729 

50271 

00021 

99979 

12 

48 

16 

49 

50108 

49892 

50130 

49870 

00022 

99978 

11 

44 

20 

50 

8.50504 

11.49496 

8.50527 

11.49473 

10.00022 

9.99978 

10 

40 

24 

51 

50897 

49103 

50920 

49080 

00023 

99977 

9 

36 

28 

52 

51287 

48713 

51310 

48690 

00023 

99977 

8 

32 

32 

53 

51673 

48327 

51696 

48304 

00023 

99977 

7 

28 

36 

54 

52055 

47945 

52079- 

47921 

00024 

99976 

6 

24 

40 

55 

8.52434 

11.47566 

8.52459 

11.47541 

10.00024 

9.99976 

5 

20 

44 

56 

52810 

47190 

52835 

47165 

00025 

99975 

4 

16 

48 

67 

53183 

46817 

53208 

46792 

00025 

99975 

O 
t j 

12 

52 

58 

53552 

46448 

53578 

46422 

00026 

99974 

2 

8 

56 

59 

53919 

46081 

53945 

46055 

00026 

99974 

1 

4 

8 

60 

54282 

45718 

54308 

45692 

00026 

99974 

0 

53 

M.S. 

6 h 

M 

91° 

Cosine. 

Secant. 

Cotangent 

Tangent. { 

Cosecant. | 

Sine. 

M 

58° 

il.S. 

5 b 

























Logarithms Trigonometric. 165 


o h 

2° 



Logarithms. 


177 c 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

[ Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

i 8 

0 

8.54282 

11.45718 

8.54308 

11.45692 

10.00026 

9.99974 

60 


4 

1 

54642 

45358 

54669 

45331 

00027 

99973 

59 

56 

8 

2 

54999 

45001 

55027 

44973 

00027 

99973 

58 

52 

12 

3 

55354 

44646 

55382 

44618 

00028 

99972 

57 

48 

16 

4 

55705 

44295 

55734 

44266 

00028 

99972 

56 

44 

20 

5 

8.56054 

11.43946 

8.56083 

11.43917 

10.00029 

9.99971 

55 

40 

24 

6 

56400 

43600 

56429 

43571 

00029 

99971 

54 

36 

28 

7 

56743 

43257 

56773 

43227 

00030 

99970 

53 

32 

32 

8 

57084 

42916 

57114 

42886 

00030 

99970 

52 

28 

36 

9 

57421 

42579 

57452 

42548 

00031 

99969 

51 

24 

40 

10 

8.57757 

11.42243 

8.57788 

11.42212 

10.00031 

9.99969 

50 

20 

44 

11 

58089 

41911 

58121 

41879 

00032 

99968 

49 

16 

48 

12 

58419 

41581 

58451 

41549 

00032 

. 99968 

48 

12 

52 

13 

58747 

41253 

58779 

41221 

00033 

99967 

47 

8 

56 

14 

59072 

40928 

59105 

40895 

00033 

99967 

46 

4 

9 

15 

8.59395 

11.40605 

8.59428 

11.40572 

10.00033 

9.99967 

45 

51 

4 

16 

59715 

40285 

59749 

40251 

00034 

99966 

44 

56 

8 

17 

60033 

39967 

60068 

39932 

00034 

99966 

43 

52 

12 

18 

60349 

39651 

60384 

39616 

00035 

99965 

42 

48 

16 

19 

60662 

39338 

60698 

39302 

00036 

99964 

41 

44 

20 

20 

8.60973 

11.39027 

8.61009 

11.38991 

10.00036 

9.99964 

40 

40 

24 

21 

61282 

38718 

61319 

38681 

00037 

99963 

39 

36 

28 

r 22 

615S9 

38411 

61626 

38374 

00037 

99963 

38 

32 

32 

23 

61894 

38106 

61931 

38069 

00038 

99962 

37 

28 

36 

24 

62196 

37804 

62234 

37766 

00038 

99962 

36 

24 

40 

25 

8.62497 

11.37503 

8.62535 

11.37465 

10.00039 

9.99961 

35 

20 

44 

26 

62795 

37205 

62834 

37166 

00039 

99961 

34 

16 

48 

27 

63091 

36909 

63131 

36869 

00040 

99960 

33 

12 

52 

28 

63385 

36615 

63426 

36574 

00040 

99960 

32 

8 

56 

29 

63678 

36322 

63718 

36282 

00041 

99959 

31 

4 

10 

30 

8.63968 

11.36032 

8.64009 

11.35991 

10.00041 

9.99959 

30 

50 

4 

31 

64256 

35744 

64298 

35702 

00042 

99958 

29 

56 

8 

32 

64543 

35457 

64585 

35415 

00042 

99958 

28 

52 

12 

33 

64827 

35173 

64870 

35130 

00043 

99957 

27 

48 

16 

34 

65110 

34890 

65154 

34846 

00044 

99956 

26 

44 

20 

35 

8.65391 

11.34609 

8.65435 

11.34565 

10.00044 

9.99956 

25 

40 

24 

36 

65670 

34330 

65715 

34285 

00045 

99955 

24 

36 

28 

37 

65947 

34053 

65993 

34007 

00045 

99955 

23 

32 

32 

38 

66223 

33777 

66269 

33731 

00046 

99954 

22 

28 

36 

39 

66497 

33503 

66543 1 

33457 

00046 

99954 

21 

24 

40 

40 

8.667 69 

11.33231 

8.66816 

11.33184 

10.00047 

9.99953 

20 

20 

44 

41 

67039 

32961 

67087 

32913 

00048 

99952 

19 

16 

48 

42 

67308 

32692 

67356 

32644 

00048 

99952 

18 

12 

' 52 

43 

67575 

32425 

67624 

32376 

00049 

99951 

17 

8 

56 

44 

67841 

32159 

67890 

32110 

00049 

99951 

16 

4 

11 

45 

8.68104 

11.31896 

8.68154 

11.31846 

10.00050 

9.99950 

15 

49 

4 

46 

68367 

31633 

68417 

31583 

00051 

99949 

14 

56 

8 

47 

68627 

31373 

68678 

31322 

00051 

99949 

13 

52 

12 

48 

68886 

31114 

68938 

31062 

00052 

99948 

12 

48~- 

16 

49 

69144 

30856 

69196 

30804 

00052 

99948 

11 

44 

20 

50 

8.69400 

11.30600 

8.69453 

11.30547 

10.00053 

9.99947 

10 

40 

24 

51 

69654 

30346 

69708 

30292 

00054 

99946 

9 

36 

28 

52 

69907 

30093 

69962 

30038 

00054 

99946 

8 

32 

32 

53 

70159 

29841 

70214 

29786 

00055 

99945 

7 

28 

36 

54 

70409 

29591 

70465 

29535 

00056 

99944 

6 

24 

40 

55 

8.70658 

11.29342 

8.70714 

11.29286 

10.00056 

9.99944 

5 

20 

44 

56 

70905 

29095 

70962 

29038 

00057 

99943 

4 

16 

48 

57 

71151 

28849 

71208 

28792 

00058 

99942 

3 

12 

52 

58 

71395 

28605 

71453 

28547 

00058 

99942 

2 

8 

56 

59 

71638 

28362 

71697 

28303 

00059 

99941 

1 

4 

13 

60 

71880 

28120 

71940 

28060 

00060 

99940 

0 

48 

M. S. 
6 h 

M 

92° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

87° 

M.S. 

5 h 






























166 Logarithms Trigonometric. 


o h 

CO 

O 



Logarithms. 


176° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

13 

0 

8.71880. 

11.28120 

8.71940 

11.28060 

10.00060 

9.99940 

60 

48 

4 

1 

72120 

27880 

72181 

27819 

00060 

99940 

59 

56 

8 

2 

72359 

27641 

72420 

27580 

00061 

99939 

58 

52 

12 

3 

72597 

27403 

72659 

27341 

000G2 

99938 

57 

48 

16 

4 

72834 

27166 

72896 

27104 

00062 

99938 

56 

44 

20 

5 

8.73069 

11.26931 

8.73132 

11.26868 

10.00063 

9.99937 

55 

40 

24 

6 

73303 

26697 

73366 

26634 

00064 

99936 

54 

36 

28 

7 

73535 

26465 

73600 

26400 

00064 

99936 

53 

32 

32 

8 

73767 

26233 

73832 

26168 

00065 

99935 

52 

28 

36 

9 

73997 

26003 

74063 

25937 

00066 

99934 

51 

24 

40 

10 

8.74226 

11.25774 

8.74292 

11.25708 

10.00066 

9.99934 

50 

20 

44 

11 

74454 

25546 

74521 

25479 

00067 

99933 

49 

16 

48 

12 

74680 • 

25320 

74748 

25252 

00068 

99932 

48 

12 

62 

13 

74906 

25094 

74974 

25026 

0006S 

99932 

47 

8 

56 

14 

75130 

24870 

75199 

24801 

00069 

99931 

46 

4 

13 

15 

8.75353 

11.24647 

8.75423 

11.24577 

10.00070 

9.99930 

45 

47 

4 

16 

75575 

24425 

75645 

24355 

00071 

99929 

44 

56 

8 

17 

75795 

24205 

75867 

24133 

00071 

99929 

43 

52 

12 

18 

76015 

23985 

76087 

23913 

00072 

99928 

42 

48 

16 

19 

76234 

23766 

76306 

23694 

00073 

99927 

41 

44 

20 

20 

8.76451 

11.23549 

8.76525 

11.23475 

10.00074 

9.99926 

40 

40 

24 

21 

76667 

23333 

76742 

23258 

00074 

99926 

39 

36 

28 

22 

76883 

23117 

76958 

23042 

00075 

99925 

38 ' 

32 

32 

23 

77097 

22903 

77173 

22827 

00076 

99924 

37 

28 

36 

24 

77310 

22690 

77387 

22613 

00077 

99923 

36 

24 

40 

25 

8.77522 

11.22478 

8.77600 

11.22400 

10.00077 

9.99923 

35 

20 

44 

26 

77733 

22267 

77811 

22189 

00078 

99922 

34 

16 

48 

27 

77943 

22057 

78022 

21978 

00079 

99921 

33 

12 

52 

28 

78152 

21848 

78232 

21768 

00080 

99920 

32 

8 

56 

29 

78360 

21640 

78441 

21559 

00080 

99920 

31 

4 

11 

30 

8.78568 

11.21432 

8.78649 

11.21351 

10.00081 

9.99919 

30 

46 

4 

31 

78774 

21226 

78855 

21145 

00082 

99918 

29 

56 

8 

32 

78979 

21021 

79061 

20939 

00083 

99917 

28 

52 

12 

33 

79183 

20817 

79266 

20734 

00083 

99917 

27 

48 

16 

34 

79386 

20614 

79470 

20530 

00084 

99916 

26 

44 

20 

35 

8.79588 

11.20412 

8.79673 

11.20327 

10.00085 

9.99915 

25 

40 

24 

36 

79789 

20211 

79875 

20125 

000S6 

99914 

24 

36 

28 

37 

79990 

200] 0 

80076 

19924 

00087 

99913 

23 

32 

32 

38 

80189 

19811 

80277 

19723 

00087 

99913 

22 

28 

36 

39 

80388 

19612 

80476 

19524 

0008S 

99912 

21 

24 

40 

40 

8.80585 

11.19415 

8.80674 

11.19326 

10.00089 

9.99911 

20 

20 

44 

41 

80782 

19218 

80872 

19128 

00090 

99910 

19 

16 

48 

42 

80978 

19022 

81068 

18932 

00091 

99909 

18 

12 

52 

43 

81173 

18827 

81264 

18736 

00091 

99909 

17 

8 

56 

44 

81367 

18633 

81459 

18541 

00092 

99908 

16 

4 

15 

45 

8.81560 

11.18440 

8.81653 

11.18347 

10.00093 

9.99907 

15 

45 

4 

46 

S1752 

18248 

81846 

18154 

00094 

99906 

14 

56 

8 

47 

81944 

18056 

82038 

17962 

00095 

99905 

13 

52 

12 

48 

82134 

17866 

82230 

17770 

00096 

99904 

12 

48 

16 

49 

82324 

17676 

82420 

17580 

00096 

99904 

11 

44 

20 

50 

8.82513 

11.17487 

8.82610 

11.17390 

10.00097 

9.99903 

10 

40 

24 

51 

82701 

17299 

82799 

17201 

00098 

99902 

9 

36 

28 

52 

82888 

17112 

82987 

17013 

00099 

99901 

8 

32 

32 

53 

83075 

16925 

83175 

16825 

00100 

99900 

7 

28 

36 

54 

83261 

16739 

83361 

16639 

00101 

99899 

6 

24 

40 

55 

8.83446 

11.16554 

8.83547 

11.16453 

10.00102 

9.99898 

5 

20 

44 

56 

83630 

16370 

83732 

16268 

00102 

99898 

4 

16 

48 

57 

83813 

16187 

83916 

160S4 

00103 

99897 

3 

12 

52 

58 

83996 

16004 

84100 

15900 

00104 

99896 

2 

8 

56 

59 

84177 

15823 

84282 

15718 

00105 

99895 

i 

4 

16 

60 

84358 

15642 

84464 

15536 

00106 

99894 

0 

44 

M.S. 

6 h 

M 

93° 

1 Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

80° 

M.S. 























Logarithms Trigonometric. 167 


0” 

4° 



Logarithms. 


175° 

IP 

M.S. 

M 

Sine. 

Cosecant. 

Taugent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S 

16 

0 

8.84358 

11.15642 

8.84464 

11.15536 

10.00106 

9.99894 

60 

•11 

4 

1 

84539 

15461 

84646 

15354 

00107 

99893 

59 

56 

8 

2 

84718 

15282 

84826 

15174 

00108 

99892 

58 

52 

12 

3 

84897 

15103 

85006 

14994 

00109 

99891 

57 

48 

16 

4 

85075 

14925 

85185 

14815 

00109 

99891 

56 

44 

20 

5 

8.85252 

11.14748 

8.85363 

11.14637 

10.00110 

9.99890 

55 

40 

24 

6 

85429 

14571 

85540 

14460 

00111 

99889 

54 

36 

28 

7 

85605 

14395 

85717 

14283 

00112 

99888 

53 

32 

32 

8 

85780 

14220 

85893 

14107 

00113 

99887 

52 

28 

36 

9 

85955 

14045 

86069 

13931 

00114 

99886 

51 

24 

40 

10 

8.86128 

11.13872 

8.86243 

11.13757 

10.00115 

9.99885 

50 

20 

44 

11 

86301 

13699 

86417 

13583 

00116 

99884 

49 

16 

48 

12 

86474 

13526 

86591 

13409 

00117 

99883 

48 

12 

62 

13 

86645 

13355 

80763 

13237 

00H8 

99882 

47 

8 

56 

14 

86816 

13184 

86935 

13065 

00119 

99881 

46 

4 

17 

15 

8.86987 

11.13013 

8.87106* 

11.12894 

10.00120 

9.99880 

45 

43 

4 

16 

87156 

12844 

87277 

12723 

00121 

99879 

44 

56 

8 

17 

87325 

12675 

87447 

12553 

00121 

99S79 

43 

52 

12 

18 

87494 

12506 

87616 

12384 

00122 

99878 

42 

48 

16 

19 

87661 

12339 

87785 

12215 

00123 

99877 

41 

44 

20 

20 

8.87829 

11.12171 

8.87953 

11.12047 

10.00124 

9.99876 

40 

40 

24 

21 

87995 

12005 

88120 

11880 

00125 

99875 

39 

36 

28 

22 

88161 

11839 

88287 

11713 

00126 

99874 

38 

32 

32 

23 

88326 

11674 

88453 

11547 

00127 

99573 

37 

28 

36 

24 

88490 

11510 

88618 

11382 

00128 

99872 

36 

24 

40 

25 

8.88664 

11.11346 

8.88783 

11.11217 

10.00129 

9.99871 

35 

20 

44 

26 

88S17 

11183 

88948 

11052 

00130 

99870 

34 

16 

48 

27 

88980 

71020 

89111 

10889 

00131 

99869 

33 

12 

52 

28 

89142 

10858 

89274 

10726 

00132 

99S68 

32 

8 

56 

29 

89304 

10696 

89437 

10563 

00133 

99S67 

31 

4 

IS 

30 

8.89464 

11.10536 

8.89598 

11.10402 

10.00134 

9.99866 

30 

12 

4 

31 

89625 

10375 

89760 

10240 

00135 

99865 

29 

56 

8 

32 

897S4 

10216 

89920 

10080 

00136 

99S64 

28 

52 

12 

33 

89943 

10057 

900S0 

09920 

00137 

99S63 

27 

48 

16 

34 

90102 

09S98 

90240 

09760 

00138 

99S62 

26 

44 

20 

35 

8.90260 

11.09740 

8.90399 

11.09601 

10.00139 

9.99861 

25 

40 

24 

36 

90417 

09583 

90557 

09443 

00140 

99S60 

24 

36 

28 

37 

90574 

09426 

90715 

09285 

00141 

99859 

23 

32 

32 

38 

90730 

09270 

90872 

09128 

00142 

99858 

22 

28 

36 

39 

90885 

09115 

91029 

08971 

00143 

99S57 

21 

24 

40 

40 

8.91040 

11.08960 

8.91185 

11.08815 

10.00144 

9.99856 

20 

20 

44 

41 

91195 

08805 

■ 91340 

08660 

00145 

99855 

19 

16 

48 

42 

91349 

08651 

91495 

08505 

00146 

99854 

18 

12 

52 

43 

91502 

08498 

91650 

08350 

00147 

99853 

17 

8 

56 

44 

91655 

08345 

91803 

08197 

00148 

99852 

16 

4 

19 

45 

8.91807 

11.08193 

8.91957 

11.08043 

10.00149 

9.99851 

15 

41 

4 

46 

91959 

08047 

92110 

07890 

00150 

99850 

14 

56 

8 

47 

92110 

07S90 

92262 

07738 

00152 

99848 

13 

52 

12 

48 

92261 

07739 

92414 

07586 

00153 

99847 

12 

48 

16 

49 

92411 

07589 

92565 

07435 

00154 

99846 

11 

44 

20 

50 

8.92561 

11.07439 

8.92716 

11.07284 

10.00155 

9.99845 

10 

40 

24 

51 

92710 

07290 

92866 

07134 

00156 

99844 

9 

36 

28 

52 

92859 

07141 

93016 

06984 

00157 

99843 

8 

32 

32 

53 

93007 

06993 

93165 

06835 

00158 

99842 

7 

28 

36 

54 

93154 

06846 

93313 

06687 

00159 

99841 

6 

24 

40 

55 

8.93301 

11.06699 

8.93462 

11.06538 

10.00160 

9.99840 

5 

20 

44 

56 

93448 

06552 

93609 

06391 

00161 

99839 

4 

16 

48 

57 

93594 

06406 

93756 

06244 

00162 

99838 

3 

12 

52 

58 

93740 

06260 

93903 

06097 

00163 

99S37 

2 

8. 

56 

59 

93885 

06115 

94049 

05951 

00164 

99836 

1 

4 

20 

60 

94030 

05970 

94195 

05805 

00166 

99834 

0 

40 

M.S. 

6 h 

M 

94° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

85° 

M.S. 

5 h 

















168 


Logarithms Trigonometric. 


0 b 

5° 



Logarithms. 


174 c 

11* 

Jl.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

so 

0 

8.94030 

11.05970 

8.94195 

11.05805 

10.00166 

9.99834 

60 

40 

4 

1 

94174 

05826 

94340 

05660 

00167 

99833 

59 

56 

8 

2 

94317 

05683 

94485 

05515 

00168 

99832 

58 

52 

12 

3 

94461 

05539 

94630 

05370 

00169 

99831 

57 

48 

16 

4 

94603 

05397 

94773 

05227 

00170 

99830 

56 

44 

20 

5 

8.94746 

11.05254 

8,94917 

11.05083 

10.00171 

9.99829 

55 

40 

24 

6 

94887 

05113 

95060 

04940 

00172 

99828 

54 

36 

28 

7 

95029 

04971 

95202 

04798 

00173 

99827 

53 

32 

32 

8 

95170 

04830 

95344 

04656 

00175 

99825 

52 

28 

36 

9 

95310 

04690 

95486 

04514 

00176 

99824 

51 

24 

40 

10 

8.95450 

11.04550 

8.95627 

11.04373 

10.00177 

9.99823 

50 

20 

44 

11 

95589 

04411 

95767 

04233 

00178 

99822 

49 

16 

48 

12 

95728 

04272 

95908 

04092 

00179 

99821 

48 

12 

52 

13 

95867 

04133 

96047 

03953 

00180 

99820 

47 

8 

56 

14 

96005 

03995 

96187 

j03813 

00181 

99819 

46 

4 

SI 

15 

8.96143 

11.03857 

8.96325 

11.03675 

10.00183 

9.99817 

45 

39 

4 

16 

96280 

03720 

96464 

03536 

00184 

99816 

44 

56 

8 

17 

96417 

03583 

96602 

03398 

00185 

99815 

43 

52 

12 

18 

96553 

03447 

96739 

03261 

00186 

99814 

42 

48 

16 

19 

96689 

03311 

96877 

03123 

00187 

99813 

41 

44 

20 

20 

8.96825 

11.03175 

8.97013 

11.02987 

10.00188 

9.99812 

40 

40 

24 

21 

96960 

03040 

97150 

02850 

00190 

99810 

39 

36 

28 

22 

97095 

02905 

97285 

02715 

00191 

99809 

38 

32 

32 

23 

97229 

02771 

97421 

02579 

00192 

9980S 

37 

28 

36 

24 

97363 

02637 

97556 

02444 

00193 

99807 

36 

24 

40 

25 

8.97496 

11.02504 

8.97691 

11.02309 

10.00194 

9.99806 

35 

20 

44 

26 

97629 

02371 

97825 

02175 

00196 

99804 

34 

16 

48 

27 

97762 

02238 

97959 

02041 

00197 

99803 

33 

12 

52 

28 

97894 

02106 

98092 

01908 

00198 

99802 

32 

8 

56 

29 

98026 

01974 

98225 

01775 

00199 

99801 

31 

4 

as 

30 

8.98157 

11.01843 

8.98358 

11.01642 

10.00200 

9.99800 

30 

38 

4 

31 

98288 

01712 

98490 

01510 

00202 

99708 

29 

56 

8 

32 

98419 

01581 

98622 

01378 

00203 

99797 

28 

52 

12 

33 

98549 

01451 

98753 

01247 

00204 

99796 

27 

48 

16 

34 

98679 

01321 

98884 

01116 

00205 

99795 

26 

44 

20 

35 

8.98808 

11.01192 

8.99015 

11.00985 

10.00207 

9.99793 

25 

40 

24 

36 

98937 

01063 

99145 

00855 

00208 

99792 

24 

36 

28 

37 

99066 

00934 

99275 

00725 

00209 

99791 

23 

32 

32 

38 

99194 

00806 

99405 

00595 

.00210 

99790 

22 

28 

36 

39 

99322 

00678 

99534 

00466 

00212 

99788 

21 

24 

40 

40 

8.99450 

11.00550 

8.99662 

11.00338 

10.00213 

9.99787 

20 

20 

44 

41 

99577 

00423 

99791 

00209 

00214 

99786 

19 

16 

48 

42 

99704 

00296 

99919 

00081 

00215 

99785 

18 

12 

52 

43 

99S30 

00170 

9.00046 

10.99954 

00217 

99783 

17 

8 

56 

44 

99956 

00044 

00174 

99826 

00218 

99782 

16 

4 

S3 

45 

9.00082 

10.99918 

9.00301 

10.99699 

10.00219 

9.99781 

15 

37 

4 

46 

00207 

99793 

00427 

99573 

00220 

99780 

14 

56 

8 

47 

00332 

99668 

00653 

99447 

00222 

99778 

13 

52 

12 

48 

00456 

99544 

00679 

99321 

00223 

99777 

12 

48 

16 

49 

00581 

99419 

00805 

99195 

00224 

99776 

11 

44 

20 

50 

9.00701 

10.99296 

9.00930 

10.99070 

10.00225 

9.99775 

10 

10 

24 

51 

00828 

99172 

01055 

98945 

00227 

99773 

9 

36 

28 

52 

00951 

99049 

01179 

98821 

00228 

99772 

8 

32 

32 

53 

01074 

98926 

01303 

98697 

00229 

99771 

7 

28 

36 

54 

01196 

98804 

01427 

98573 

00231 

99769 

6 

24 

40 

55 

9.01318 

10.98682 

9.01550 

10.98450 

10.00232 

9.99768 

5 

20 

44 

56 

01440 

98560 

01673 

98327 

00233 

99767 

4 

16 

48 

57 

01561 

98439 

01796 

98204 

00235 

99765 

3 

12 

' 52 

58 

01682 

9S31S 

01918 

98082 

00236 

99764 

2 

8 

56 

59 

01803 

98197 

02040 

97960 

00237 

99763 

1 

4 

24 

60 

01923 

98077 

02162 

97838 

00239 

99761 

0 

3G 

M.S. 

6 h 

M 

95° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

B4° 

M. S. 

5“ 





















Logarithms Trigonometric. 169 



6° 



Logarithms. 


173° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

34 

0 

9.01923 

10.98077 

9.02162 

10.97838 

10.00239 

9.9970 L 

GO 

36 

4 

i 

02043 

97957 

02283 

97717 

00240 

99760 

59 

56 

8 

2 

02163 

97837 

02404 

97596 

00241 

99759 

58 

52 

12 

3 

02283 

97717 

02525 

97475 

00243 

99757 

57 

48 

16 

4 

02402 

97598 

02645 

97355 

00244 

99756 

56 

44 

20 

5 

9.02520 

10.97480 

9.02766 

10.97234 

10.00245 

9.99755 

55 

40 

24 

6 

02639 

97361 

02885 

97115 

00247 

99753 

54 

36 

28 

7 

02757 

97243 

03005 

96995 

00248 

99752 

53 

32 

32 

8 

02874 

97126 

03124 

96876 

00249 

99751 

52 

28 

36 

9 

02992 

97008 

03242 

96758 

00251 

99749 

51 

24 

40 

10 

9.03109 

10.96891 

9.03361 

10.96639 

10.00252 

9.99748 

50 

20 

44 

11 

03226 

96774 

03479 

96521 

00253 

99747 

49 

16 

48 

12 

03342 

96658 

03597 

96403 

00255 

99745 

48 

12 

52 

13 

03458 

96542 

03714 

96286 

00256 

99744 

47 

8 

56 

14 

03574 

96420 

03832 

96168 

00258 

99742 

46 

4 

35 

15 

9.03690 

10.96310 

9.03948 

10.96052 

10.00259 

9.99741 

45 

35 

4 

16 

03805 

96195 

04065 

95935 

00260 

99740 

44 

55 

8 

17 

03920 

96080 

04181 

95819 

00262 

99738 

43 

52 

12 

18 

04034 

95966 

04297 

95703 

00263 

99737 

42 

48 

16 

19 

04149 

95851 

04413 

95587 

00264 

99736 

41 

44 

20 

20 

9.04262 

10.95738 

9.04528 

10.95472 

10.00266 

9.99731 

40 

40 

24 

21 

04376 

95624 

04643 

95357 

00267 

99733 

39 

36 

28 

22 

04490 

95510 

04758 

95242 

00269 

99731 

38 

32 

32 

23 

04603 

95397 

04873 

95127 

00270 

99730 

37 

28 

36 

24 

04715 

95285 

04987 

95013 

00272 

99728 

35 

24 

40 

25 

9.04828 

10.95172 

9.05101 

10.94899 

10.00273 

9.99727 

35 

20 

44 

26 

04940 

95060 

05214 

94786 

00274 

99726 

34 

16 

48 

27 

05052 

94948 

05328 

94672. 

00276 

99724 

33 

12 

52 

28 

05164 

94836 

05441 

94559 

00277 

99723 

32 

8 

56 

29 

05275 

94725 

05553 

94447 

00279 

99721 

31 

4 

36 

30 

9.05386 

10.94614 

9.05666 

10.94334 

10.00280 

9.99720 

30 

34 

4 

31 

05497 

94503 

05778 

94222 

00282 

99718 

29 

55 

8 

32 

05607 

94393 

05890 

94110 

00283 

99717 

28 

52 

12 

33 

05717 

94283 

06002 

93998 

00284 

99716 

27 

48 

16 

34 

05827 

94173 

06113 

93887 

00286 

99714 

26 

41 

20 

35 

9.05937 

10.94063 

9.06224 

10.93776 

10.00287 

9.99713 

25 

40 

24 

36 

06046 

93954 

06335 

93665 

00289 

99711 

24 

35 

28 

37 

06155 

93845 

06445 

93555 

00290 

99710 

23 

32 

82 

38 

06264 

93736 

06556 

93444 

00292 

99708 

22 

28 

36 

39 

06372 

93628 

06666 

93334 

00293 

99707 

21 

24 

40 

40 

9.06481 

' 10.93519 

9.06775 

10.93225 

10.00295 

9.99705 

20 

20 

44 

41 

06589 

93411 

06885 

93115 

00296 

99704 

19 

16 

48 

42 

06696 

93304 

06994 

93006 

00298 

99702 

18 

12 

52 

43 

06804 

93196 

07103 

92897 

00299 

99701 

17 

8 

56 

44 

06911 

93089 

07211 

92789 

00301 

97699 

16 

4 

37 

45 

9.07018 

10.92982 

9.07320 

10.92680 

10.00302 

9.99698 

15 

33 

4 

46 

07124 

92876 

07428 

92572 

00304 

99696 

14 

56 

8 

47 

07231 

92769 

07536 

92464 

00305 

99695 

13 

52 

12 

48 

07337 

92663 

07643 

92357 

00307 

99693 

12 

48 

16 

49 

07442 

92558 

07751 

92249 

00308 

99692 

11 

44 

20 

50 

9.07548 

10.92452 

9.07858 

10.92142 

10.00310 

3.99690 

10 

40 

24 

51 

07 653 

92347 

07964 

92036 

00311 

99689 

9 

36 

28 

52 

07758 

92242 

08071 

91929 

00313 

99687 

8. 

32 

32 

53 

07863 

92137 

08177 

91823 

00314 

99686 

7 

28 

36 

54 

07968 

92032 

08283 

91717 

00316 

99684 

6 

24 

40 

55 

9.08072 

i 0.91928 

9.08389 

10.91611 

10.00317 

9.99683 

5 

20 

44 

56 

08176 

91824 

08495 

91505 

00319 

99681 

4 

16 

48 

57 

08280 

91720 

08600 

91400 

00320 

99680 

3 

12 

52 

58 

0S383 

91617 

08705 

91295 

00322 

99678 

2 

8 

56 

59 

08486 

91514 

08810 

91190 

00323 

99677 

1 

4 

38 

60 

08589 . 

91411 

08914 

91086 

00325 

99675 

0 

33 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

m i 

M.S. 

6 U 

96° 







83°| 





















170 Logarithms Trigonometric. 


o h 

7° 



Logarithms. 


172° 

ll h 

M. S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

| Secant. 

Cosiue. 

M 

M.S. 

28 

0 

9.08589 

10.91411 

9.08914 

10.91086 

10.00325 

9.9J675 

60 

325 

' 4 

1 

08692 

91308 

09019 

90981 

00326 

99674 

59 

56 

8 

2 

08795 

91205 

09123 

90877 

00328 

99672 

58 

52 

12 

3 

08897 

91103 

09227 

90773 

00330 

99670 

57 

48 

16 

4 

08999 

91001 

09330 

90670 

00331 

99669 

56 

41 

20 

5 

9.09101 

10.90899 

9.09434 

10.90566 

10.00333 

9.99667 

55 

40 

24 

6 

09202 

90798 

09537 

90463 

00334 

99666 

54 

36 

28 

7 

09304 

90096 

09640 

90360 

00336 

99664 

53 

32 

32 

8 

09405 

90595 

09742 

90258 

00337 

99663 

52 

28 

30 

9 

09506 

90494 

09845 

90155 

00339 

99661 

51 

24 

40 

10 

9.09606 

10.90394 

9.09947 

10.90053 

10.00341 

9.99659 

50 

20 

44 

11 

09707 

90293 

10049 

89951 

00342 

99658 

49 

16 

48 

12 

09807 

90193 

10150 

89850 

00344 

99656 

48 

12 

52 

13 

09907 

90093 

10252 

89748 

00345 

99655 

47 

8 

50 

14 

10006 

89994 

10353 

89647 

00347 

99653 

46 

4 

39 

15 

9.10106 

10.89894 

9.10454 

10.89546 

10.00349 

9.99651 

45 

31 

4 

16 

10205 

89795 

10555 

89445 

00350 

99650 

44 

56 

8 

17 

10304 

89696 

10656 

89344 

00352 

9964S 

43 

52 

12 

18 

10402 

89598 

10756 

89244 

00353 

99647 

42 

48 

10 

19 

10501 

89499 

10856 

89144 

00355 

99645 

41 

44 

20 

20 

9.10599 

10.89401 

9.10956 

10.89044 

10.00357 

9.99643 

40 

40 

24 

21 

10697 

89303 

11056 

8S944 

00358 

99642 

39 

36 

28 

22 

10795 

89205 

11155 

88845 

00360 

99640 

38 

32 

32 

23 

10893 

89107 

11254 

88746 

00362 

99638 

37 

28 

30 

24 

10990 

89010 

11353 

88647 

00363 

99637 

36 

24 

40 

25 

9.11087 

10.88913 

9.11452 

10.88548 

10.00305 

9.99635 

35 

20 

44 

26 

11184 

88816 

11551 

88449 

00367 

99633 

34 

16 

48 

27 

11281 

88719 

11649 

88351 

00368 

99632 

33 

12 

52 

28 

11377 

88623 

11747 

88253 

00370 

99630 

32 

8 

56 

29 

11474 

88526 

11845 

88155 

00371 

99629 

31 

4 

30 

30 

9.11570 

10.88430 

9.11943 

10.88057 

10.00373 

9.99627 

30 

30 

4 

31 

11666 

88334 

12040 

87960 

00375 

99625 

29 

56 

8 

32 

11761 

88239 

12138 

87862 

00376 

99624 

28 

52 

12 

33 

11857 

88143 

12235 

87765 

00378 

99622 

27 

48 

16 

34 

11952 

8S048 

12332 

87668 

00380 

99620 

26 

44 

20 

35 

9.12047 

10.87953 

9.12428 

10.87572 

10.00382 

9.99618 

25 

40 

24 

36 

12142 

87858 

12525 

87475 

00383 

99617 

24 

36 

28 

37 

12236 

87764 

12621 

87379 

003S5 

99615 

23 

32 

32 

3S 

12331 

87669 

12717 

87283 

00387 

99613 

22 

28 

30 

39 

12425 

87575 

12813 

87187 

00388 

99612 

21 

24 

40 

40 

9.12519 

10.87481 

9.12909 

10.87091 

10.00390 

- 9.99610 

20 

20 

44 

41 

12012 

87388 

13004 

86996 

00392 

99608 

19 

16 

48 

42 

12706 

87294 

13099 

86901 

00393 

99607 

18 

12 

52 

43 

12799 

87201 

13194 

86806 

00395 

99605 

17 

8 

56 

44 

12892 

87108 

13289 

86711 

00397 

99603 

16 

4 

31 

45 

9.12985 

10.87015 

9.13384 

10.86616 

10.00399 

9.99601 

15 

29 

4 

46 

13078 

86922 

13478 

86522 

00400 

99600 

14 

56 

8 

47 

13171 

86829 

13573 

86427 

00402 

99598 

13 

62 

12 

48 

13263 

86737 

13667 

86333 

00404 

99596 

12 

48 

10 

49 

13355 

86645 

13761 

86239 

00405 

99595 

11 

44 

20 

50 

9.13447 

10.86553 

9.13854 

10.86146 

10.00407 

9.99593 

10 

40 

24 

51 

13539 

86461 

13948 

86052 

00409 

99591 

9 

36 

28 

52 

13630 

86370 

14041 

85959 

00411 

99589 

8 

32 

32 

53 

13722 

86278 

14134 

85866 

00412 

99588 

7 

28 

36 

54 

13S13 

86187 

14227 

85773 

00414 

99586 

6 

24 

40 

55 

9.13904 

10.86096 

9.14320 

10.85680 

10.00416 

9.99584 

5 

20 

44 

56 

13994 

86006 

14112 

85588 

0041S 

995S2 

4 

16 

48 

57 

14085 

85915 

14504 

85496 

00419 

99581 

3 

12 

52 

58 

14175 

85825 

14597 

85403 

00421 

99579 

2 

8 

50 

59 

14206 

85734 

14688 

85312 

00423 

99577 

1 

4 

33 

60 

14356 

85644 

14780 

85220 

00425 

. 99576 

o 

28 

M.S. 

6 h 

M 

97° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

82°| 

M.S. 

5 fa 

























Logarithms Trigonometric. 171 


0 h 

GO 

O 



Logarithms. 


171° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

33 

0 

9.14356 

10.85644 

9.14780 

10.85220 

10.00125 

9.99575 

60 

38 

4 

1 

14445 

85555 

14872 

85128 

00426 

99574 

59 

56 

8 

2 

14535 

85465 

14963 

85037 

0042S 

99572 

58 

52 

12 

3 

14624 

85376 

15054 

84946 

00430 

99570 

57 

48 

16 

4 

14714 

85286 

15145 

84855 

00432 

99568 

56 

44 

20 

5 

9.14803 

10.85197 

9.15236 

10.84764 

10.00434 

9.99566 

55 

40 

24 

6 

14891 

85109 

15327 

84673 

00435 

99565 

54 

36 

28 

7 

14980 

8502a 

15417 

84583 

00437 

99563 

53 

32 

32 

8 

15069 

84931 

15508 

84492 

00439 

99561 

52 

28 

36 

9 

15157 

84843 

15598 

84402 

00441 

99559 

51 

24 

40 

10 

9.15245 

10.84755 

9.15688 

10.84312 

10.00443 

9.99557 

50 

20 

44 

11 

15333 

84667 

15777 

84223 

00444 

99556 

49 

16 

48 

12 

15421 

84579 

15867 

84133 

00446 

99554 

48 

12 

52 

13 

15508 

84492 

15956 

84044 

00448 

99552 

47 

8 

56 

14 

15596 

84404 

16046 

83954 

00450 

99550 

46 

4 

33 

15 

9.15683 

10.84317 

9.16135 

10.83865 

10.00452 

9.99548 

45 

37 

4 

16 

15770 

84230 

16224 

83776 

00454 

99546 

44 

56 

8 

17 

35857 

84143 

16312 

83688 

00455 

99545 

43 

52 

12 

18 

15944 

84056 

16401 

83599 

00457 

99543 

42 

48 

16 

19 

16030 

83970 

16489 

83511 

00459 

99541 

41 

44 

20 

20 

9.16116 

10.83884 

9.16577 

10.83423 

10.00461 

9.99539 

40 

40 

24 

21 

16203 

83797 

16665 

83335 

00463 

99537 

39 

36 

28 

22 

16289 

83711 

16753 

83247 

00465 

99535 

38 

32 

32 

23 

16374 

83626 

16841 

83159 

.00467 

99533 

37 

28 

36 

24 

16460 

S3540 

16928 

83072 

00468 

99532 

36 

24 

40 

25 

9.16545 

10.83455 

9.17016 

10.82984 

10.00470 

9.99530 

35 

20 

44 

26 

16631 

83369 

17103 

82897 

00472 

99528 

34 

16 

48 

27 

16716 

83284 

17190 

82810 

00474 

99526 

33 

12 

52 

28 

16S01 

83199 

17277 

82723 

00476 

99524 

32 

8 

56 

29 

16886 

83114 

17363 

82637 

00478 

99522 

31 

4 

34 

30 

9.16970 

10.83030 

9.17450 

10.82550 

10.00480 

9.99520 

30 

36 

4 

31 

17055 

82945 

17536 

82464 

00482 

99518 

29 

56 

8 

32 

17139 

82861 

17622 

82378 

00483 

99517 

28 

52 

12 

33 

17223 

82777 

17708 

82292 

00485 

99515 

27 

4S 

16 

34 

17307 

82693 

17794 

82206 

00487 

99513 

26 

44 

20 

35 

9.17391 

10.82609 

9.17880 

10.82120 

10.00489 

9.99511 

25 

40 

24 

36 

17474 

82526 

17965 

82035 

00491 

99509 

24 

36 

28 

37 

17558 

82442 

18051 

81949 

00493 

99507 

23 

32 

32 

38 

17641 

82359 

18136 

81864 

00495 

99505 

22 

28 

36 

39 

17724 

82276 

18221 

81779 

00497 

99503 

21 

24 

40 

40 

9.17807 

10.82193 

9.18306 

10.81694 

10.00499 

0.99501 

20 

20 

44 

41 

17890 

82110 

18391 

81609 

00501 

99499 

19 

16 

48 

42 

17973 

82027 

18475 

81525 

00503 

99497 

18 

12 

52 

43 

18055 

81945 

18560 

81440 

00505 

99495 

17 

8 

56 

44 

18137 

81863 

18644 

81356 

00506 

99494 

16 

4 

35 

45 

9.18220 

10.81780 

9.18728 

10.81272 

10.00508 

9.99492 

15 

35 

4 

46 

18302 

81698 

18812 

81188 

00510 

99490 

14 

56 

8 

47 

18383 

81617 

18896 

81104 

00512 

99488 

13 

52 

12 

48 

18465 

81535 

18979 

81021 

00514 

99486 

12 

48 

16 

49 

18547 

81453 

19063 

80937 

00516 

99484 

11 

44 

20 

50 

9.18628 

10.81372 

9.19146 

10.80854 

10.00518 

9.99482 

10 

40 

24 

51 

18709 

81291 

19229 

80771 

00520 

99480 

9 

36 

28 

52 

18790 

81210 

19312 

80688 

00522 

99178 

8 

32 

32 

53 

18871 

81129 

19395 

80605 

00524 

99476 

7 

28 

36 

54 

18952 

81048 

19478 

80522 

00526 

99474 

6 

24 

40 

55 

9.19033 

10.80967 

9.19561 

10.80439 

10.00528 

9.99472 

5 

20 

44 

56 

19113 

808S7 

19643 

80357 

00530 

99470 

4 

16 

48 

57 

19193 

80807 

19725 

80275 

00532 

99468 

3 

12 

52 

58 

19273 

80727 

19807 

80193 

00534 

99466 

2 

8 

56 

59 

19353 

80647 

19889 

80111 

00536 

99464 

1 

4 

30 

60 

19433 

80567 

19971 

80029 

00538 

99462 

0 

34 

M.S. 

6 h 

M 

98° 

Cosiue. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

81° 

M. S. 

5 h 

















172 


Logarithms Trigonometric. 


0* 

O 



Logarithms. 


170 c 

11“ 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

36 

0 

9.19433 

10.80567 

9.19971 

10.80029 

10.00538 

9.99462 

60 

31 

4 

1 

19513 

80487 

20053 

79947 

00540 

99460 

59 

56 

8 

2 

19592 

80408 

20134 

79866 

00542 

99458 

58 

52 

12 

3 

19672 

80328 

20216 

79784 

00544 

99466 

57 

48 

10 

4 

19751 

80249 

20297 

79703 

00546 

99454 

56 

44 

20 

5 

9.19830 

10.80170 

9.20378 

10.79622 

10.00548 

9.99452 

55 

40 

24 

6 

19909 

80091 

20459 

79541 

00550 

99450 

54 

SB 

28 

7 

19988 

80012 

20540 

79460 

. 00552 

99448 

53 

32 

82 

8 

20067 

79933 

20621 

79379 

00554 

99446 

52 

28 

36 

9 

20145 

79855 

20701 

79299 

00556 

99444 

51 

24 

40 

Hi 

9.20223 

10.79777 

9.20782 

10.79218 

10.00558 

9.99442 

50 

20 

44 

11 

20802 

79698 

20862 

79138 

00560 

99440 

49 

16 

48 

12 

20380 

79620 

20942 

79058 

00562 

99438 

48 

12 

52 

13 

20458 

79542 

21022 

78978 

00564 

99436 

47 

8 

56 

14 

20535 

79465 

21102 

7S898 

00566 

99434 

46 

4 

37 

15 

9.20613 

10.79387 

9.21182 

10.78818 

10.00568 

9.99432 

45 

£3 

4 

16 

20691 

79309 

21261 

78739 

00571 

99429 

44 

56 

8 

17 

20768 

79232 

21341 

78659 

00573 

99427 

43 

52 

12 

18 

20845 

79155 

21420 

78580 

00575 

99425 

42 

48 

16 

19 

20922 

79078 

21499 

78501 

00577 

99423 

41 

44 

20 

20 

9.20999 

10.79001 

9.21578 

10.78422 

10.00579 

9.99421 

40 

40 

24 

21 

21076 

78924 

21657 

78343 

00581 

99419 

39 

36 

28 

22 

21153 

78847 

21736 

78264 

00583 

99417 

38 

32 

32 

23 

21229 

78771 

21814 

78186 

00585 

99415 

37 

28 

3G 

24 

21306 

78694 

21893 

78107 

00587 

99413 

36 

24 

40 

25 

9.21382 

10.78618 

9.21971 

10.78029 

10.00589 

9.99411 

35 

20 

44 

26 

21458 

78542 

22049 

77951 

00591 

99409 

34 

16 

48 

27 

21534 

78466 

22127 

77873 

00593 

99407 

33 

12 

52 

28 

21610 

78390 

22205 

77795 

00596 

99404 

32 

8 

56 

29 

21685 

78315 

22283 

77717 

00598 

99402 

31 

4 

38 

30 

9.21761 

10.78239 

9.22361 

10.77639 

10.00600 

9.99400 

30 

22 

4 

•31 

21836 

78164 

22438 

77562 

00602 

99398 

29 

56 

8 

32 

21912 

78088 

22516 

77484 

00604 

99396 

28 

52 

12 

33 

21987 

78013 

22593 

77407 

00606 

99394 

27 

48 

16 

34 

22062 

77938 

22670 

77330 

0(1608 

99392 

26 

44 

20 

35 

9.22137 

10.77863 

9.22747 

10.77253 

10.00610 

9.99390 

25 

40 

24 

36 

22211 

77789 

22824 

77176 

00612 

99388 

24 

36 

28 

37 

22286 

77714 

229(11 

77099 

00615 

99385 

23 

32 

32 

38 

22361 

77639 

22977 

77023 

00617 

99383 

22 

28 

36 

39 

22435 

77565 

23054 

76946 

00619 

99381 

21 

24 

40 

40 

9.22509 

10.77491 

9.23130 

10.76870 

10.00621 

9.99379 

20 

20 

44 

41 

22583 

77417 

23206 

76794 

00623 

99377 

19 

16 

4S 

42 

22657 

77343 

23283 

76717 

00625 

99375 

18 

12 

52 

43 

22731 

77269 

23359 

76641 

00628 

99372 

17 

8 

56 

44 

22805 

77195 

23135 

76565 

00630 

99370 

16 

4 

36 

45 

9.22878 

10.77122 

9.23510 

10.76490 

10.00632 

9.99368 

15 

21 

4 

46 

22952 

77048 

23586 

76414 

00634 

99366 

14 

56 

8 

47 

23025 

76975 

23661 

76339 

00636 

99364 

13 

52 

12 

48 

23098 

76902 

23737 

76263 

00638 

99362 

12 

48 

16 

49 

23171 

76829 

23812 

76188 

00641 

99359 

11 

44 

20 

50 

9.23244 

10.76756 

9.23887 

10.76113 

10.00643 

9.99357 

10 

40 

24 

51 

23317 

76683 

23962 

76038 

00645 

99355 

9 

36 

28 

52 

23390 

76610 

24037 

75963 

00647 

99353 

8 

32 

32 

53 

23462 

76538 

24112 

75888 

00649 

99351 

7 

28 

36 

54 

23535 

76465 

24186 

75S14 

00652 

99348 

6 

24 

40 

55 

9.23607 

10.76393 

9.24261 

10.75739 

10.00654 

9.99346 

5 

20 

44 

56 

23679 

76321 

24335 

75665 

00656 

99344 

4 

16 

48 

57 

23752 

76248 

24410 

75590 

00658 

99342 

3 

12 

52 

58 

23823 

76177 

24484 

75516 

00660 

99340 

2 

8 

56 

59 

23895 

76105 

24558 

75442 

00663 

99337 

1 

4 

40 

60 

23967 

76033 

24632 

75368 

00665 

99336 

0 

20 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent, l 

Cosecant. 

Sine. 

M 

M. S. 

6 h 

99° 






oc 

O 

o 

5 h 



















Logarithms Trigonometric. 173 


o h 

10° 



Logarithms. 


169° 

ll h 

M.S. 

* M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

| M 

M.S. 

40 

o 

9.23967 

10.76033 

9.24632 

10.75368 

10.00665 

9.99335 

60 

£0 

4 

1 

24039 

75961 

24706 

75294 

00667 

99333 

59 

56 

8 

2 

24110 

75890 

24779 

75221 

00669 

99331 

58 

52 

12 

3 

24181 

75819 

24853 

75147 

00672 

99328 

57 

48 

16 

4 

24253 

75747 

24926 

75074 

00674 

99326 

56 

44 

20 

5 

9.24324 

10.75676 

9.25000 

10.75000 

10.00676 

9.99324 

55 

40 

24 

6 

24395 

75605 

25073 

74927 

00678 

99322 

54 

36 

28 

7 

24466 

75534 

25146 

74854 

00681 

99319 

53 

32 

32 

8 

24536 

75464 

25219 

74781 

00683 

99317 

52 

28 

36 

9 

24607 

75393 

25292 

74708 

00685 

99315 

51 

24 

40 

10 

9.24677 

10.75323 

9.25365 

10.74635 

10.00687 

9.99313 

50 

20 

44 

11 

24748 

75252 

25437 

74563 

00690 

99310 

49 

16 

48 

12 

24818 

75182 

25510 

74490 

00692 

99308 

48 

12 

52 

13 

24888 

75112 

25582 

74418 

00694 

99306 

47 

8 

56 

14 

24958 

75042 

25655 

74345 

00696 

99304 

46 

4 

41 

15 

9.25028 

10.74972 

9.25727 

10.74273 

10.00699 

9.99301 

45 

19 

4 

16 

25098 

74902 

25799 

74201 

00701 

99299 

44 

56 

8 

17 

25168 

74832 

25871 

74129 

00703 

99297 

43 

52 

12 

L8 

25237 

74763 

25943 

74057 

00706 

99294 

42 

48 

16 

19 

25307 

74693 

26015 

73985 

00708 

99292 

41 

44 

20 

20 

9.25376 

10.74(524 

9.26086 

10.73914' 

10.00710 

9.99290 

40 

40 

24 

21 

25445 

74555 

26158 

73842 

00712 

99288 

39 

36 

j 28 

22 

25514 

74486 

26229 

73771 

00715 

99285 

38 

32 

j 32 

23 

25583 

74417 

26301 

73699 

00717 

99283 

37 

28 

36 

24 

25652 

74348 

26372 

73628 

00719 

99281 

36 

24 

40 

25 

9.25721 

10.74279 

9.26443 

10.73557 

10.00722 

9.99278 

35 

20 

44 

26 

25790 

74210 

26514 

73486 

00724 

99276 

34 

16 

48 

27 

25858 

74142 

26585 

73415 

00726 

99274 

33 

12 

52 

28 

25927 

74073 

26655 

73345 

00729 

99271 

32 

8 

56 

29 

25995 

74005 

26726 

73274 

00731 

99269 

31 

4 

4£ 

30 

9.26063 

10.73937 

9.26797 

10.73203 

10.00733 

9.99267 

30 

18 

4 

31 

2^)131 

73869 

26867 

73133 

00736 

99264 

29 

56 

8 

32 

26199 

73801 

26937 

73063 

00738 

99262 

28 

52 

12 

33 

26267 

73733 

27008 

72992 

00740 

99260 

27 

48 

16 

34 

26335 

73665 

27078 

72922 

00743 

99257 

26 

44 

20 

35 

9.26403 

10.73597 

9.27148 

10.72852 

10.00745 

9.99255 

25 

40 

24 

36 

26470 

73530 

27218 

72782 

00748 

99252 

24 

36 

28 

37 

26538 

73462 

27288 

72712 

00750 

99250 

23 

32 

32 

38 

26605 

73395 

27357 

72643 

00752 

99248 

22 

28 

36 

39 

26672 

73328 

27427 

72573 

00755 

99245 

21 

24 

40 

40 

9.26739 

10.73261 

9.27496 

10.72504 

10.00757 

9.99243 

20 

20 

44 

41 

26806 

73194 

27566 

72434 

00759 

99241 

19 

16 

48 

42 

26873 

73127 

27635 

72365 

00762 

99238 

18 

12 

52 

43 

26940 

73060 

27704 

72296 

00764 

99236 

17 

8 

56 

44 

27007 

72993 

27773 

72227 

00767 

99233 

16 

4 

43 

45 

9.27073 

10.72927 

9.27842 

10.72158 

10.00769 

9.99231 

15 

17 

4 

46 

27140 

72860 

27911 

72089 

00771 

99229 

14 

56 

8 

47 

27206 

72794 

27980 

72020 

00774 

99226 

13 

52 

12 

48 

27273 

72727 

28049 

71951 

00776 

99224 

12 

48 

16 

49 

27339 

72661 

28117 

71883 

00779 

99221 

11 

44 

20 

50 

9.27405 

10.72595 

9.28186 

10.71814 

10.00781 

9.99219 

10 

40 

24 

51 

27471 

72529 

28254 

71748 

00783 

99217 

9 

36 

28 

52 

27537 

72463 

28323 

71677 

00786 

99214 

8 

32 

32 

53 

27602 

72398 

28391 

71609 

00788 

99212 

7 

28 

36 

54 

27663 

72332 

28459 

71541 

00791 

99209 

6 

24 

40 

55 

9.27734 

10.72266 

9.28527 

• 10.71473 

10.00793 

9.99207 

5 

20 

44 

56 

27799 

72201 

28595 

71405 

00796 

99204 

4 

16 

48 

57 

27864 

72136 

28662 

71338 

00798 

99202 

3 

12 

52 

58 

27930 

72070 

2872.0 

71270 

00800 

99200 

2 

8 

56 

59 

27995 

72005 

28798 

71202 

00803 

99197 

1 

4 

44 

60 

28060 

71940 

28S65 

71135 

00805 

99195 

0 

1« 

M.S. 

6 h 

M 

100' 

Cosine. 1 
D 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

79° 

M.S. 

5 h 



























274 Logarithms Trigonometric. 


o h 

11° 



Logarithms. 


168° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

ir 

M.S. 

44 

0 

9.28060 

10.71940 

9.28865 

10.71135 

10.00805 

9.99195 


1G 

4 

1 

28125 

71875 

28933 

71067 

00808 

99192 

59 

56 

8 

2 

28190 

71810 

29000 

71000 

00810 

99190 

58 

52 

12 

3 

28254 

71746 

29067 

70933 

00S13 

99137 

57 

48 

15 

4 

28319 

71681 

29134 

70806 

00815 

99185 

56 

41 

20 

5 

9.28384 

10.71616 

9.29201 

10.70799 

10.00818 

9.99182 

55 

40 

24 

6 

28448 

71552 

29268 

70732 

00820 

99180 

54 

36 

28 

7 

28512 

71488 

29335 

70665 

00823 

99177 

53 

32 

32 

8 

28577 

71423 

29402 

70598 

00825 

99175 

52 

28 

30 

9 

28641 

71359 

29408 

70532 

00828 

99172 

51 

24 

40 

10 

9.28705 

10.71295 

•9.29535 

10.70465 

10.00830 

9.99 L70 

50 

20 

44 

11 

28769 

71231 

29601 

70399 

00S33 

99167 

49 

16 

48 

12 

28833 

71167 

2966S 

70332 

00835 

99165 

48 

12 

52 

13 

28S96 

71104 

29734 

70266 

00838 

99162 

47 

8 

58 

14 

28960 

71040 

29800 

70200 

00S40 

99160 

46 

4 

45 

15 

9.29024 

10.70976 

9.29866 

10.70134 

10.00843 

9.99157 

45 

15 

4 

16 

29087 

70913 

29932 

70068 

00845 

99155 

44 

56 

8 

17 

29150 

' 70850 

29998 

70002 

00818 

99152 

43 

52 

12 

18 

29214 

70786 

30064 

69936 

00850 

99150 

42 

48 

16 

19 

29277 

70723 

30130 

6987G 

00853 

99147 

41 

44 

20 

20 

9.29340 

10.70660 

9.30195 

10.69805 

10.00855 

9.99145 

40 

40 

24 

21 

29403 

70597 

30201 

69739 

00858 

99142 

39 

36 

28 

22 

29466 

70534 

30326 

69674 

00860 

99140 

3S 

32 

32 

23 

29529 

70471 

30391 

69609 

00863 

99137 

37 

28 

36 

24 

29591 

70409 

30457 

69543 

00865 

99135 

36 

24 

40 

25 

9.29654 

10.70346 

9.30522 

10.69478 

10.00868 

9.99132 

35 

20 

44 

26 

29716 

70284 

30587 

69413 

00870 

99130 

34 

16 

48 

27 

29779 

70221 

30652 

69348 

00873 

99127 

33 

12 

52 

28 

29841 

70159 

30717 

69283 

00376 

99124 

32 

8 

56 

29 

29903 

70097 

30782 

69218 

00878 

99122 

31 

4 

46 

30 

9.29966 

10.70034 

9.30846 

10.69154 

10.00881 

9.99119 

30 

1-4 

4 

31 

30028 

69972 

30911 

69089 

00883 

99117 

29 

56 

8 

32 

30090 

69910 

30975 

69025 

00886 

99114 

28 

52 

12 

33 

3U151 

69849 

31040 

68960 

00888 

99112 

27 

48 

16 

34 

30213 

697S7 

31104 

68S96 

00891 

99109 

26 

44 

20 

35 

9.30275 

10.69725 

9.31168 

10.68832 

10.00894 

9.99106 

25 

40 

24 

36 

30336 

69664 

31233 

68767 

00896 

99104 

24 

36 

28 

37 

30398 

69602 

31297 

68703 

00899 

99101 

23 

32 

32 

38 

30459 

69541 

31361 

68639 

00901 

99099 

22 

28 

36 

39 

30521 

69479 

31425 

68575 

00904 

99096 

21 

24 

40 

40 

9.30582 

10.69418 

9.31489 

10.68511 

10.00907 

9.99093 

20 

20 

44 

41 

30643 

69357 

31552 

68448 

00909 

99091 

19 

16 

48 

42 

30704 

69296 

31616 

68384 

00912 

99088 

18 

12 

52 

43 

30765 

69235 

31679 

68321 

00914 

99086 

17 

8 

56 

44 

30826 

69174 

31743 

68257 

00917 

99083 

16 

4 

47 

45 

9.30887 

10.69113 

9.31806 

10.68194 

10.00920 

9.99080 

15 

13 

4 

46 

30947 

69053 

31870 

68130 

00922 

99078 

14 

5G 

8 

47 

31008 

68992 

31933 

68067 

00925 

99075 

13 

52 

12 

48 

31068 

68932 

31996 

68004 

00928 

99072 

12 

48 

16 

49 

31029 

68871 

32059 

67941 

00930 

99070 

11 

44 

20 

50 

9.31189 

10.68811 

9.32122 

10.67878 

10.00933 

9.99067 

10 

40 

24 

51 

31250 

68750 

32185 

67815 

00936 

99064 

9 

36 

28 

52 

31310 

68690 

32248 

67752 

00938 

99062 

8 

32 

32 

53 

31370 

68630 

32311 

67689 

00941 

99059 

7 

28 

36 

54 

31430 

68570 

32373 

67627 

00944 

99056 

6 

24 

40 

55 

9.31490 

10.68510 

9.32430 

10.67564 

10.00946 

9.99054 

5 

20 

44 

56 

31549 

68451 

32498 

67502 

00949 

99051 

4 

16 

48 

57 

31609 

68391 

32561 

67439 

00952 

99048 

3 

12 

52 

58 

31669 

68331 

32623 

67377 

00954 

99046 

2 

8 

5G 

59 

31728 

68272 

32685 

67315 

00957 

99043 

1 

4 

48 

60 

31788 

68212 

32747 

67253 

00960 

99040 

0 

lti 

M.S. 

6 h 

M 

101 

Cosiue. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

78° 

M.S. 

5» 



























Logarithms Trigonometric. 175 


0 b 

12° 



Logarithms. 


167° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

48 

0 

9.31788 

10.68212 

9.32747 

10.67253 

10.00960 

9.99040 

60 

13 

4 

1 

31847 

68153 

32810 

67190 

00962 

99038 

59 

56 

8 

2 

31907 

68093 

32872 

67128 

00965 

99035 

58 

52 

12 

3 

31966 

68034 

32933 

67067 

00968 

99032 

57 

48 

16 

4 

32025 

67975 

32995 

67005 

00970 

99030 

56 

44 

20 

5 

9.32084 

10.67916 

9.33057 

10.66943 

10.00973 

9.99027 

55 

40 

24 

6 

32143 

67857 

33119 

66881 

00976 

96024 

54 

36 

28 

7 

32202 

67798 

33180 

66820 

00978 

99022 

53 

32 

32 

8' 

32261 

67739 

33242 

66758 

00981 

99019 

52 

28 

36 

9 

32319 

67681 

33303 

66697 

00984 

99016 

51 

24 

40 

10 

9.32378 

10.67622 

9.33365 

10.66635 

10.00987 

9.99013 

50 

20 

44 

11 

32437 

67563 

33426 

66574 

00989 

99011 

49 

16 

48 

12 

32495 

67505 

33487 

66513 

00992 

99008 

48 

12 

52 

13 

32553 

67447 

33548 

66452 

00995 

99005 

47 

8 

56 

14 

32612 

67388 

33609 

66391 

00998 

99002 

46 

4 

49 

15 

9.32670 

10.67330 

9.33670 

10.66330 

10.01000 

9.99000 

45 

11 

4 

16 

32728 

67272 

33731 

66269 

01003 

98997 

44 

56 

8 

17 

32786 

67214 

33792 

66208 

01006 

98994 

43 

52 

12 

18 

32844 

67156 

33853 

66147 

01009 

98991 

42 

48 

16 

19 

32902 

67098 

33913 

66087 

01011 

98989 

41 

44 

20 

20 

9.32960 

10.67040 

9.33974 

10.60026 

10.01014 

9.98986 

40 

40 

24 

21 

33018 

66982 

34034 

65966 

01017 

98983 

39 

36 

28 

22 

33075 

66925 

34095 

65905 

01020 

98980 

3S 

32 

32 

23 

33133 

66867 

34155 

65845 

01022 

98978 

37 

28 

36 

24 

33190 

66810 

34215 

65785 

01025 

98975 

36 

24 

40 

25 

9.33248 

10.66752 

9.34276 

10.65724 

10.01028 

9.98972 

35 

20 

44 

26 

33305 

66695 

34336 

65664 

01031 

98969 

34 

16 

48 

27 

33362 

66638 

34396 

65604 

01033 

98967 

33 

12 

52 

28 

33420 

66580 

34456 

65544 

01036 

98964 

32 

8 

56 

29 

33477 

66523 

34516 

65484 

01039 

98961 

31 

4 

50 

30 

9.33534 

10.66466 

9.34576 

10.65424 

10.01042 

9.98958 

30 

10 

4 

31 

33591 

66409 

34635 

65365 

01045 

98955 

29 

56 

8 

32 

33647 

66353 

34695 

65305 

01047 

98953 

28 

52 

12 

33 

33704 

66296 

34755 

65245 

01050 

98950 

27 

48 

16 

34 

33761 

66239 

' 34814 

65186 

01053 

98947 

26 

44 

20 

35 

9.33818 

10.66182 

9.34874 

10.65126 

10.01056 

9.98944 

25 

40 

24 

36 

33874 

66126 

34933 

65067 

01059 

98941 

24 

36 

28 

37 

33931 

66069 

34992 

65008 

01062 

98938 

23 

32 

32 

38 

33987 

66013 

35051 

64949 

01064 

98936 

22 

28 

36 

39 

34043 

65957 

35111 

64889 

01067 

98933 

21 

24 

40 

40 

9.34100 

10.65900 

9.35170 

10.64830 

10.01070 

9.98930 

20 

20 

44 

41 

34156 

65844 

35229 

64771 

01073 

98927 

19 

16 

48 

42 

34212 

65788 

35288 

64712 

01076 

98924 

18 

12 

52 

43 

34268 

65732 

35347 

64653 

01079 

98921 

17 

8 

56 

44 

34324 

65676 

35405 

64595 

01081 

98919 

16 

4 

51 

45 

9.34380 

10.65620 

9.35404 

10.64536 

10.01084 

9.98916 

15 

O 

4 

46 

34436 

65564 

35523 

64477 

01087 

98913 

14 

56 

8 

47 

34491 

65509 

35581 

64419 

01090 

98910 

13 

52 

12 

48 

34547 

65453 

35640 

64360 

01093 

98907 

12 

48 

16 

49 

34602 

65398 

35698 

64302 

01096 

,98904 

11 

44 

20 

50 

9.34658 

10.65342 

9.35757 

10.64243 

10.01099 

9.98901 

10 

40 

24 

51 

34713 

65287 

35815 

64185 

01102 

98898 

9 

36 

28 

52 

34769 

65231 

35873 

64127 

01104 

98896 

8 

32 

32 

53 

34824 

65176 

35931 

64069 

01107 

98893 

7 

28 

36 

54 

34879 

65121 

35989 

64011 

OHIO 

98890 

6 

24 

40 

55 

9.34934 

10 65066 

9.36047 

10.63953 

10.01113 

9.98887 

5 

20 

44 

56 

34989 

65011 

36105 

63895 

01116 

98884 

4 

16 

48 

57 

35044 

64950 

36163 

63837 

01119 

98881 

3 

12 

52 

58 

35099 

64901 

36221 

63779 

01122 

98878 

2 

8 

56 

59 

35154 

64846 

36279 

63721 

01125 

98375 

1 

4 

52 

60 

35209 

64791 

36336 

63664 

01128 

98872 

0 

8 

M.S. 

6 h 

M 

102 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

77° 

M.S. 

5“ 






















176 Logarithms Trigonometric. 


1 

0 h 

13 c 



Logarithms. 


166 c 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

53 

0 

9.35209 

10.64791 

9.36336 

10.63604 

10.01128 

9.98872 

60 

8 

4 

1 

35263 

64737 

36394 

63606 

01131 

98869 

59 

56 

8 

2 

35318 

64682 

36452 

63548 

01133 

98867 

58 

52 

12 

ft 

O 

35373 

64627 

36509 

63491 

01136 

98864 

57 

48 

1G 

4 

35427 

64573 

36566 

63434 

01139 

98861 

56 

44 

20 

5 

9.35481 

10.64519 

9.36624 

10.63376 

10.01142 

9.98858 

55 

40 

24 

6 

35536 

64464 

36681 

63319 

01145 

98855 

54 

36 

28 

7 

35590 

64410 

36738 

63262 

01148 

98852 

53 

32 

32 

8 

35644 

64356 

36795 

63205 

01151 

98849 

52 

28 

3G 

9 

35698 

64302 

36852 

63148 

01154 

98846 

51 

24 

40 

10 

9.35752 

10.64248 

9.36909 

10.63091 

10.01157 

9.98843 

50 

20 

44 

11 

35806 

64194 

36966 

63034 

01160 

98840 

49 

16 

48 

12 

35860 

64140 

37023 

62977 

01163 

9S837 

48 

12 

52 

13 

35914 

64086 

37080 

62920 

01166 

98834 

47 

8 

56 

14 

35968 

64032 

37137 

62863 

01169 

98831 

46 

4 

53 

15 

9.36022 

10.63978 

9.37193 

10.62807 

10.01172 

9.98828 

45 

7 

4 

16 

36075 

63925 

37250 

62750 

01175 

98825 

44 

56 

8 

17 

36129 

63871 

37306 

62694 

01178 

98822 

43 

52 

12 

18 

36182 

63818 

37363 

62637 

01181 

98819 

42 

48 

1G 

19 

36236 

63764 

37419 

62581 

01184 

98816 

41 

44 

20 

20 

9.36289 

10.63711 

9.37476 

10.62524 

10.01187 

9.98813 

40 

40 

24 

21 

36342 

63658 

37532 

62468 

01190 

98810 

39 

36 

28 

22 

36395 

63605 

37588 

62412 

01193 

98807 

38 

32 

32 

23 

36449 

63551 

37G44 

62356 

01196 

98804 

37 

28 

3G 

24 

36502 

63498 

37700 

62300 

01199 

98801 

36 

24 

40 

25 

9.36555 

10.63445 

9.37756 

10.62244 

10.01202 

9.98798 

35 

20 

44 

26 

36608 

63392 

37812 

62188 

01205 

98795 

34 

16 

48 

27 

36660 

63340 

37868 

62132 

01208 

98792 

33 

12 

52 

28 

36713 

63287 

37924 

62076 

01211 

98789 

32 

8 

5G 

29 

36766 

63234 

37980 

62020 

01214 

98786 

31 

4 

54: 

30 

9.36819 

10.63181 

9.38035 

10.61965 

10.01217 

9.98783 

30 

G 

4 

31 

36871 

63129 

38091 

61909 

01220 

98780 

29 

56 

8 

32 

36924 

63076 

38147 

61853 

01223 

98777 

28 

52 

12 

33 

36976 

63024 

38202 

61798 

01226 

98774 

27 

48 

16 

34 

37028 

62972 

38257 

61743 

01229 

98771 

26 

44 

20 

35 

9.37081 

10.62919 

9.38313 

10.61687 

10.01232 

9.98768 

25 

40 

24 

36 

37133 

62867 

38368 

61632 

01235 

98765 

24 

36 

28 

37 

37185 

628 L5 

38423 

61577 

01238 

98762 

23 

3-2 

32 

38 

37237 

62763 

38479 

61521 

01241 

98759 

22 

28 

36 

39 

37289 

62711 

38534 

61466 

01244 

98756 

21 

24 

40 

40 

9.37341 

10.62659 

9.38589 

10.61411 

10.01247 

9.98753 

20 

20 

44 

41 

37393 

62607 

38644 

61356 

01250 

98750 

19 

16 

48 

42 

37445 

62555 

38699 

61301 

01254 

98746 

18 

12 

52 

43 

37497 

62503 

38754 

61246 

01257 

98743 

17 

8 

56 

44 

37549 

62451 

38808 

61192 

01260 

98740 

16 

4 

55 

45 

9.37600 

10.62400 

9.38863 

10.61137 

10.01263 

9.98737 

15 

5 

4 

46 

37652 

62348 

38918 

61082 

01266 

98734 

14 

56 

8 

47 

37703 

62297 

38972 

61028 

01269 

98731 

13 

52 

12 

48 

37755 

62245 

39027 

60973 

01272 

98728 

12 

48 

16 

49 

37806 

62194 

39082 

60918 

01275 

98725 

11 

44 

20 

50 

9.37858 

10.62142 

9.39136 

10.60864 

10.01278 

9.98722 

10 

40 

24 

51 

37909 

62091 

39190 

60810 

01281 

98719 

9 

36 

28 

52 

37960 

62040 

39245 

60755 

01285 

98715 

8 

32 

32 

53 

38011 

619S9 

39299 

60701 

01288 

98712 

7 

28 

36 

54 

38062 

61938 

3935 > 

60647 

01291 

98709 

6 

24 

40 

56 

9.38113 

10 61887 

9.39407 

10.60593 

10.01294 

9.98706 

5 

20 

44 

56 

38164 

61836 

3946 L 

60539 

01297 

98703 

4 

18 

48 

57 

38215 

61785 j 

39515 

60485 

01300 

98700 

3 

12 

52 

58 

38266 

61734 

39569 

60431 

01303 

98697 

2 

8 

66 

59 

38317 

61083 

39623 

60377 

01306 

986.-4 

1 

4 

50 

60 

38368 

61632 

39677 

60323 | 

01310 

98680 

0 

4: 

M.S. 

(i* 

M 

103 

Cosine. 

0 

Secant. 

Cotangent 

Tangent, i 

Cosecant. 

Sine. 

M 

70° 

M. S. 

5 h 


























Logarithms Trigonometric. 177 


o h 

14° 



Logarithms, 

• 

165° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

1 M 

M.S. 

56 

0 

9.38368 

10.61632 

9.39677 

10.60323 

10.01310 

9.98690 

60 

4 

4 

1 

38418 

61582 

39731 

60269 

01313 

98687 

59 

56 

8 

2 

38169 

61531 

39785 

60215 

01316 

98684 

58 

52 

12 

3 

38519 

61481 

39838 

60162 

01319 

98681 

57 

48 

16 

4 

38570 

61430 

39892 

60108 

01322 

98678 

56 

44 

20 

5 

9.38620 

10.61380 

9.39945 

10.60055 

10.01325 

9.98675 

55 

40 

24 

6 

38670 

61330 

39999 

60001 

01329 

98671 

54 

36 

28 

7 

38721 

61279 

40052 

59948 

01332 

98668 

53 

32 

32 

8 

38771 

61229 

40106 

59894 

01335 

98665 

52 

28 

36 

9 

38821 

61179 

40159 

59841 

01338 

98662 

51 

24 

40 

10 

9.3887 l 

10.61129 

9.40212 

10.59788 

10.01341 

9.98659 

50 

20 

44 

11 

38921 

61079 

40266 

59734 

01344 

98656 

49 

16 

48 

12 

38971 

61029 

40319 

59681 

01348 

98652 

48 

12 

52 

13 

39021 

60979 

40372 

59628 

01351 

98649 

47 

8 

56 

14 

39071 

60929 

40425 

59575 

01354 

98646 

46 

4 

57 

15 

9.39121 

10.60879 

9.40478 

10.59522 

10.01357 

9.98643 

45 

3 

4 

16 

39170 

60830 

40531 

59469 

01360 

98640 

44 

56 

8 

17 

39220 

60780 

40584 

59416 

01364 

98636 

43 

52 

12 

18 

39270 

60730 

40636 

59364 

01367 

93633 

42 

48 

16 

19 

39319 

60681 

40689 

59311 

01370 

98630 

41 

44 

20 

20 

9.39369 

10.60631 

9.40742 

10.59258 

10.01373 

9.98627 

40 

40 

24 

21 

39418 

60582 

40795 

59205 

01377 

98623 

39 

36 

28 

22 

39467 

60533 

40847 

59153 

01380 

98620 

38 

32 

32 

23 

39517 

60483 

40900 

59100 

01383 

98617 

37 

28 

36 

24 

39566 

60434 

40952 

59048 

01386 

98614 

36 

24 

40 

25 

9.39615 

10.60385 

9.41005 

10.58995 

10.01390 

9.98610 

35 

20 

44 

26 

39664 

60336 

41057 

58943 

01393 

98607 

34 

16 

48 

27 

39713 

60287 

41109 

58891 

01396 

98604 

33 

12 

52 

28 

39762 

6023S 

41161 

58839 

01399 

98601 

32 

8 

56 

29 

39811 

60189 

41214 

58786 

01403 

98597 

31 

4 

58 

30 

9.39860 

10.60140 

9.41266 

10.58734 

10.01406 

9.98594 

30 

2 

4 

31 

39909 

60091 

41318 

58682 

01409 

.98591 

29 

56 

8 

32 

39958 

60042 

41370 

58630 

01412 

98588 

28 

52 

12 

33 

40006 

59994 

41422 

58578 

01416 

98584 

27 

48 

16 

34 

40055 

59945 

41474 

58526 

01419 

98581 

26 

41 

20 

35 

9.40103 

10.59897 

9.41526 

10.58474 

10.01422 

9.98578 

25 

40 

24 

36 

40152 

59848 

41578 

58422 

01426 

98574 

24 

36 

28 

37 

40200 

59800 

41629 

58371 

01429 

98571 

23 

32 

32 

38 

40249 

59751 

41681 

58319 

01432 

98568 

22 

28 

36 

39 

40297 

59703 

41733 

58267 

01435 

98565 

21 

24 

40 

40 

9.40346 

10.59654 

9.41784 

10.58216 

10.01439 

9.98561 

20 

20 

44 

41 

40394 

59606 

41836 

58164 

01442 

98558 

19 

16 

48 

42 

40442 

59558 

41887 

58113 

01445 

98555 

18 

12 

52 

43 

40490 

59510 

41939 

58061 

01449 

98551 

17 

8 

56 

44 

40538 

59462 

41990 

58010 

01452 

98548 

16 

4 

59 

45 

9.40586 

10.59414 

9.42041 

10.57959 

10.01455 

9.98545 

15 

1 

4 

46 

40634 

59366 

42093 

57907 

01459 

98541 

14 

56 

8 

47 

40682 

59318 

42144 

57856 

01462 

98538 

13 

52 

12 

48 

40730 

59270 

42195 

57805 

01465 

98535 

12 

48 

16 

49 

40778 

59222 

42246 

57754 

01469 

98531 

11 

44 

20 

50 

9.40825 

10.59175 

9.42297 

10.57703 

10.01472 

9.98528 

10 

10 

24 

51 

40873 

59127 

42348 

57652 

01475 

98525 

9 

36 

28 

52 

40921 

59079 

42399 

57601 

01479 

98521 

8 

32 

32 

53 

40968 

59032 

42450 

57550 

01482 

98518 

7 

28 

36 

54 

41016 

58984 

42501 

57499 

01485 

98515 

6 

24 

40 

55 

9.41063 

10.58937 

9.42552 

10.57448 

10.01489 

9.98511 

5 

20 

44 

56 

41111 

58889 

42603 

57397 

01492 

98508 

4 

16 

48 

57 

41158 

58842 

42653 

57347 

01495 

98505 

3 

12 

52 

58 

41205 

58795 

42704 

57296 

01499 

98501 

2 

8 

56 

59 

41252 

58748 

42755 

57245 

01502 

98498 

1 

4 

60 

60 

41300 

58700 

42805 

57195 

01506 

98494 

0 

O 

M.S. 

6 h 

M ' 

104 c 

Cosine. 

Secant, j 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

75° 

M.S. 

5“ 


12 






















178 Logarithms Trigonometric. 


l h 

15° 


• 

Logarithms. 


164° 

10 h 

M.S. 

M 

Siue. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

0 

0 

9.41300 

10.58700 

9.42805 

10.57195 

10.01506 

9.98494 

60 

60 

4 

1 

41347 

58653 

42856 

57144 

01509 

98491 

59 

56 

8 

2 

41394 

58606 

42906 

57094 

01512 

98488 

58 

52 

12 

3 

41441 

58559 

42957 

57043 

01516 

98484 

57 

4S 

16 

4 

414S8 

58512 

43007 

56993 

01519 

98481 

56 

44 

20 

5 

9.41535 

10.58465 

9.43057 

10.56943 

10.01523 

9.98477 

55 

40 

24 

6 

41582 

58418 

43108 

56892 

01526 

98474 

54 

36 

28 

7 

41628 

58372 

43158 

56842 

01529 

98471 

53 

32 

32 

8 

41675 

58325 

43208 

56792 

01533 

98467 

52 

28 

36 

9 

41722 

58278 

43258 

56742 

01536 

98464 

51 

24 

40 

10 

9.41768 

10.58232 

9.43308 

10.56092 

10.01540 

9.98460 

50 

20 

44 

11 

41815 

58185 

43358 

56642 

01543 

98457 

49 

16 

48 

12 

41861 

58139 

43408 

56592 

01547 

98453 

48 

12 

52 

13 

41908 

58092 

43458 

56542 

01550 

98450 

47 

8 

56 

14 

41954 

58046 

43508 

56492 

01553 

98447 

46 

4 

1 

15 

9.42001 

10.57999 

9.43558 

10.56442 

10.01557 

9.98443 

45 

59 

4 

16 

42047 

57953 

43607 

56393 

01560 

98440 

44 

56 

8 

17 

42093 

57907 

43657 . 

56343 

01564 

98436 

43 

52 

12 

18 

42140 

57860 

43707 

56293 

01567 

98433 

42 

48 

16 

19 

42186 

57814 

43756 

56244 

01571 

98429 

41 

44 

20 

20 

9.42282 

10.57768 

9.43806 

10.56194 

10.01574 

9.98426 

40 

40 

24 

21 

42278 

57722 

43855 

56145 

01578 

98422 

39 

36 

28 

22 

42324 

57676 

43905 

56095 

01581 

98419 

38 

32 

32 

23 

42370 

57630 

43954 

56046 

01585 

98415 

37 

28 

36 

24 

42416 

57584 

44004 

55996 

01588 

98412 

36 

24 

40 

25 

9.42461 

10.57539 

9.44053 

10.55947 

10.01591 

9.98409 

35 

20 

44 

26 

42507 

57493 

44102 

55898 

01595 

98405 

34 

16 

48 

27 

42553 

57447 

44151 

55849 

01598 

98402 

33 

12 

52 

28 

42599 

57401 

44201 

55799 

01602 

98398 

32 

8 

56 

29 

42644 

57356 

44250 

55750 

01605 

98395 

31 

4 

3 

30 

9.42690 

10.57310 

9.44299 

10.55701 

10.01609 

9.98391 

30 

58 

4 

31 

42735 

57265 

44348 

55652 

01612 

98388 

29 

56 

8 

32 

42781 

57219 

44397 

55603 

01616 

98384 

28 

52 

12 

33 

42826 

57174 

44446 

55554 

01619 

98381 

27 

48 

16 

34 

42872 

57128 

44495 

55505 

01623 

98377 

26 

41 

20 

35 

9.42917 

10.57083 

9.44544 

10.55456 

10.01627 

9.98373 

25 

40 

24 

36 

42962 

57038 

44592 

55408 

01630 

98370 

24 

36 

28 

37 

43008 

56992 

44641 

55359 

01634 

98366 

23 

32 

32 

38 

43053 

56947 

44690 

55310 

01637 

98363 

22 

28 

36 

39 

43098 

50902 

44738 

55262 

01641 

98359 

21 

24 

40 

40 

9.43143 

10.56857 

9.44787 

10.55213 

10.01644 

9.98356 

20 

20 

44 

41 

43188 

56812 

44836 

55164 

01648 

98352 

19 

16 

48 

42 

43233 

56767 

44884 

55116 

01651 

98349 

18 

12 

52 

43 

43278 

56722 

44933 

55067 

01655 

98345 

17 

8 

56 

44 

43323 

56677 

449S1 

55019 

01658 

98342 

16 

4 

3 

45 

9.43367 

10.56633 

9.45029 

10.54971 

10.01662 

9.98338 

15 

57 

4 

46 

43412 

56588 

45078 

54922 

01666 

98334 

14 

56 

8 

47 

43457 

56543 

45126 

54874 

01669 

98331 

13 

52 

12 

48 

43502 

56498 

45174 

54826 

01673 

98327 

12 

48 

16 

49 

43546 

56454 

45222 

54778 

01676 

98324 

11 

44 

20 

50 

9.43591 

10.56409 

9.45271 

10.54729 

10.01680 

9.98320 

10 

40 

24 

51 

43635 

56365 

45319 

54681 

01683 

98317 

9 

36 

28 

52 

43680 

56320 

45367 

54633 

01687 

. 98313 

8 

32 

32 

53 

43724 

56276 

45415 

545S5 

01691 

98309 

7 

28 

36 

54 

43769 

56231 

45463 

54537 

01094 

98306 

6 

24 

40 

55 

9.43813 

10.56187 

9.45511 

10.54489 

10.01698 

9.98302 

5 

20 

44 

56 

43857 

56143 

45559 

54441 

01701 

98299 

4 

16 

48 

57 

43901 

56099 

45606 

54394 

01705 

98295 

3 

12 

52 

58 

43946 

56054 

45654 

51346 

01709 

98291 

2 

8 

56 

59 

48990 

56010 

45702 

54298 

01712 

98288 

1 

4 

4 

60 

44034 

55966 

45750 

54250 

01716 

9S284 

0 

56 

M.S. 

7 h 

M 

105 

Cosine. 

D 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

74° 

M.S. 

4 h 


















Logarithms Trigonometric. 179 


l h 

16° 



Logarithms. 


163° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

4 

0 

9.44034 

10.55966 

9.45750 

10.54250 

10.01716 

9.98284 

60 

56 

4 

1 

44078 

55922 

45797 

54203 

01719 

98281 

59 

56 

8 

o 

44 

44122 

55878 

45845 

54155 ' 

01723 

98277 

58 

52 

12 

3 

44166 

55834 

45892 

54108 

01727 

98273 

57 

48 

16 

4 

44210 

55790 

45940 

54060 

01730 

98270 

56 

44 

20 

5 

9.44253 

10.55747 

9.45087 

10.54013 

10.01734 

9.9S266 

55 

40 

24 

6 

44297 

55703 

46035 

53965 

01738 

9S262 

54 

36 

28 

7 

44341 

55659 

46082 

53918 

01741 

98259 

53 

32 

32 

8 

44385 

65615 

46130 

53S70 

01746 

98255 

52 

28 

30 

9 

44428 

55572 

46177 

53823 

01749 

98251 

51 

24 

40 

10 

9.44472 

10.55528 

9.46224 

10.53776 

10.01752 

9.98248 

50 

20 

44 

11 

44516 

55484 

46271 

53729 

01766 

98244 

-±9 

16 

48 

12 

44559 

55441 

46319 

63681 

01760 

98240 

48 

12 

52 

13 

44602 

55398 

46366 

53634 

01763 

98237 

47 

8 

56 

14 

44646 

55354 

46413 

53587 

01767 

98233 

46 

4 

5 

15 

9.44689 

10.55311 

9.46460 

10.53540 

10.01771 

9.98229 

45 

55 

4 

16 

44733 

55267 

46507 

53493 

01774 

98226 

44 

56 

8 

17 

44776 

55224 

46554 

53446 

01778 

98222 

43 

52 

12 

18 

44819 

55181 

46601 

53399 

01782 

98218 

42 

48 

16 

19 

44862 

55138 

46648 

53352 

01785 

98215 

41 

44 

20 

20 

9.44905 

10.55095 

9.46694 

10.53306 

10.01789 

9.98211 

40 

40 

24 

21 

44948 

55052 

46741 

63259 

01793 

98207 

39 

36 

28 

22 

44992 

55008 

46788 

53212 

01796 

98204 

38 

32 

32 

23 

45035 

54965 

46835 

53165 

01800 

9S200 

37 

28 

36 

24 

45077 

51923 

46881 

53119 

01804 

98196 

36 

24 

40 

25 

9.45120 

10.54880 

9.46928 

10.53072 

10.01808 

9.98192 

35 

20 

44 

26 

45163 

54837 

46975 

53025 

01811 

98189 

34 

16 

48 

27 

45206 

54794 

47021 

62979 

01815 

98185 

33 

12 

52 

28 

45249 

54751 

47068 

52931 

01819 

98181 

32 

8 

56 

29 

45292 

54708 

47114 

52886 

01823 

98177 

31 

4 

6 

30 

9.45334 

10.54666 

9.47160 

10.52840 

10.01826 

9.98174 

30 

54: 

4 

31 

45377 

54623 

47207 

52793 

01830 

98170 

29 

56 

8 

32 

45419 

54581 

47253 

62747 

01834 

98166 

28 

52 

12 

33 

45402 

54538 

47499 

52701 

01838 

98162 

27 

48 

16 

34 

45504 

54496 

47346 

52664 

01841 

98159 

26 

44 

20 

35 

9.45517 

10,54453 

9.47392 

10.52608 

10.01845 

9.98155 

25 

40 

24 

36 

45589 

54411 

47438 

52562 

01849 

98151 

24 

36 

28 

37 

45632 

54368 

47484 

52516 

01853 

98147 

23 

32 

u2 

38 

45674 

54326 

47530 

52470 

01856 

98144 

22 

28 

36 

39 

45716 

54284 

47576 

52424 

01860 

98140 

21 

24 

40 

40 

9.45758 

10.54242 

9.47622 

10.52378 

10.01861 

9.98136 

20 

20 

44 

41 

45801 

54190 

47668 

52332 

018 6> 

98132 

19 

16 

48 

42 

45843 

54157 

47714 

62286 

01871 

98129 

18 

12 

52 

43 

46885 

54115 

47760 

52240 

01875 

98125 

17 

8 

56 

44 

45927 

54073 

47806 

52194 

01879 

98121 

16 

4 

7 

45 

9.45969 

10.54031 

9.47852 

10.52148 

10.01883 

9.98117 

15 

53 

4 

46 

46011 

53980 

47897 

52103 

01887 

98113 

14 

56 

8 

47 

46053 

53947 

47943 

52057 

01890 

98110 

13* 

52 

12 

48 

46095 

53905 

47989 

52011 

01894 

98106 

12 

48 

16 

49 

46136 

53864 

48035 

51965 

01898 

98102 

11 

44 

20 

50 

9.46178 

10.53822 

9.48080 

10.51920 

10.01902 

9.98098 

10 

40 

24 

51 

46220 

53780 

48126 

51874 

01906 

98094 

9 

36 

28 

52 

16262 

53738 

48171 

51829 

01910 

98090 

8 

32 

32 

53 

46303 

53697 

48217 

51783 

01913 

98087 

7 

28 

36 

54 

40345 

53655 

48262 

51738 

01917 

98083 

6 

24 

40 

55 

9.46386 

10 53614 

9.48307 

10.51693 

10.01921 

9.98079 

5 

20 

4 4 

56 

46428 

53572 

48353 

51647 

01925 

98075 

4 

16 

48 

57 

46469 

53531 

48398 

51602 

01929 

98071 

3 

12 

52 

58 

46511 

53489 

48443 

51567 

01933 

98067 

2 

8 

56 

59 

46552 

53448 

48489 

51511 

01937 

98063 

1 

4 

8 

60 

46594 

53406 

48534 

51466 

01940 

98060 

0 

52 

M.S. 

7 h 

M 

10G 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

73° 

SI. s. 

4 h 
















180 Logarithms Trigonometric. 


l b 

17° 



Logarithms. 


162° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

8 

0 

9.46594 

10.53406 

9.48534 

10.51466 

10.01910 

9.98060 

60 

53 

4 

1 

46635 

53365 

48579 

51421 

01914 

98056 

59 

56 

8 

2 

46676 

53324 

48624 

51376 

01948 

98052 

58 

52 

12 

3 

46717 

53283 

48669 

51331 

01952 

98048 

57 

48 

16 

4 

46758 

53242 

48714 

51286 

01956 

98044 

56 

44 

20 

5 

9.46800 

10.53200 

9.48759 

10.51241 

10.01960 

9.98040 

55 

40 

24 

6 

46841 

53159 

48804 

51196 

01961 

98036 

54 

36 

28 

7 

46882 

53118 

48849 

51151 

01968 

98032 

53 

32 

32 

8 

46923 

53077 

48894 

51106 

01971 

98029 

52 

28 

36 

9 

46964 

53036 

48939 

51061 

01975 

98025 

51 

24 

40 

10 

9.47005 

10.52995 

9.48984 

10.51016 

10.01979 

9.98021 

50 

20 1 

44 

11 

47045 

52955 

49029 

50971 

01983 

98017 

49 

16 ! 

48 

12 

47086 

52914 

49073 

50927 

01987 

98013 

48 

12 

52 

13 

47127 

52873 

49118 

50882 

01991 

98009 

47 

8 

56 

14 

47168 

52832 

49163 

50837 

01995 

98005 

46 

4 

9 

15 

9.47209 

10.52791 

9.49207 

10.50793 

10.01999 

9.98001 

45 

51 

4 

16 

47249 

52751 

49252 

50748 

02003 

97997 

44 

56 

8 

17 

47290 

52710 

49296 

50704 

02007 

97993 

43 

52 

12 

18 

47330 

52670 

49341 

50659 

02011 

97989 

42 

4S 

16 

19 

47371 

52629 

49385 

50615 

02014 

97986 

41 

44 

20 

20 

9.47411 

10.52589 

9.49430 

10.50570 

10.02018 

9.97982 

40 

40 

24 

21 

47452 

52548 

49474 

50526 

02022 

9797S 

39 

36 

28 

22 

47492 

52508 

49519 

50481 

02026 

97974 

38 

32 

32 

23 

47533 

52467 

49563 

50437 

02030 

97970 

37 

28 

36 

24 

47573 

52427 

49607 

50393 

02031 

97966 

36 

24 

40 

25 

9.47613 

10.52387 

9.49652 

10.50348 

10.02038 

9.97962 

35 

20 

44 

26 

47654 

52346 

49696 

50304 

02042 

97958 

34 

16 

48 

27 

47694 

52306 

49740 

50260 

02046 

97954 

33 

12 

52 

28 

47734 

52266 

49784 

50216 

02050 

97950 

32 

8 

56 

29 

47774 

52226 

49828 

50172 

02054 

97946 

31 

4 

10 

30 

9.47814 

10.52186 

9.49872 

10.50128 

10.02058 

9.97942 

30 

50 

4 

31 

47854 

52146 

49916 

50084 

02062 

9793S 

29 

56 

8 

32 

47894 

52106 

49960 

50040 

02066 

97934 

2S 

52 

12 

33 

47934 

52066 

50004 

49996 

02070 

97930 

27 

48 

16 

34 

47974 

52026 

50'-48 

49952 

02074 

97926 

26 

44 

20 

35 

9.48014 

10.51986 

9.50092 

10.49908 

10.02078 

9.97922 

25 

40 

24 

36 

48054 

51946 

50136 

49864 

02082 

97918 

24 

36 

28 

37 

48094 

51906 

50180 

49820 

02086 

97914 

23 

32 

32 

38 

48133 

51867 

50223 

49777 

02090 

97910 

22 

28 

36 

39 

48173 

51827 

50267 

49733 

02094 

97906 

21 

24 

40 

40 

9.48213 

10.51787 

9.50311 

10.49689 

10.02098 

9.97902 

20 

20 

44 

41 

48252 

51748 

50355 

49645 

02102 

97898 

19 

16 

48 

42 

48292 

51708 

50398 

49602 

02106 

97894 

18 

12 

52 

43 

48332 

51668 

50442 

49558 

02110 

97890 

17 

8 

56 

44 

48371 

51629 

60485 

49515 

02114 

97886 

16 

4 

11 

45 

9.48411 

10.51589 

9.50529 

10.49471 

10.02118 

9.97882 

15 

■40 

4 

46 

48450 

51550 

50572 

4942 s 

02122 

97878 

14 

56 

8 

•47 

48490 

51510 

50616 

49384 

02126 

97874 

13 

52 

12 

48 

48529 

51471 

50659 

49341 

02130 

97870 

12 

48 1 

16 

49 

48568 

51432 

50703 

49297 

02134 

97866 

11 

44 

20 

50 

9.48607 

10.51393 

9.50746 

10.49254 

10.02139 

9.97861 

10 

40 

24 

51 

48647 

51353 

50789 

•49211 

02143 

97857 

9 

36 

28 

52 

48686 

51314 

50833 

49167 

02147 

97853 

8 

32 

32 

53 

48725 

51275 

50876 

49124 

02151 

97849 

7 

28 

36 

54 

4S764 

51236 

50919 

49081 

02155 

97845 

6 

24 

40 

55 

9.48803 

1051197 

9.50962 

10.49038 

10.02159 

9.97841 

5 

20 

44 

56 

48842 

51158 

51005 

48995 

02163 

97837 

'4 

16 

48 

57 

48881 

51119 

51048 

48952 

02167 

97833 

3 

12 

52 

58 

48920 

51080 

51092 

48908 

02171 

97829 

2 

8 

56 

59 

48959 

51041 

51135 

48865 

02175 

97825 

1 

4 

13 

60 

48998 

51002 

51178 

48822 

02179 

97821 

0 

•48 

M. 8. 

?h 

M 

107 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

700 

1 4nJ 

M. S. 

4 b 




























Logarithms Tiugonomktric. 181 


1“ 

18° 



IiOi-urll Intis. 


o 

r—i 

10“ 

M.S, 

M 

Slue. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

Vi 

0 

9.48998 

10.51002 

9.61178 

10.48822 

10.02179 

9.97821 

00 

48 

4 

1 

49037 

50903 

51221 

48779 

02183 

97817 

59 

50 

8 

2 

49070 

60024 

61204 

48730 

02188 

97812 

58 

52 

Vi 

3 

49115 

50885 

51300 

48094 

02192 

97808 

57 

48 

its 

4 

49153 

60847 

61349 

48051 

02196 

97804 

60 

44 

20 

6 

9.49192 

10.50808 

9.51392 

10.48608 

10.02200 

9.97800 

55 

40 

24 

0 

4923 L 

50709 

61435 

48565 

02204 

97790 

54 

36 

28 

7 

49209 

50731 

61478 

48522 

02208 

97792 

53 

32 

32 

8 

49308 

50092 

61520 

48480 

02212 

97788 

52 

28 

30 

9 

49347 

50063 

51503 

48437 

02216 

97784 

51 

24 

40 

10 

9.49385 

10.50015 

9.51006 

10.48394 

10.02221 

9.97779 

60 

20 

44 

11 

49424 

50570 

61048 

48352 

02225 

97775 

49 

10 

48 

12 

49402 

5053S 

51691 

48309 

02229 

97771 

48 

12 

62 

13 

49500 

50600 

51734 

48266 

02233 

97707 

47 

8 

r.ti 

14 

49539 

60401 

51770 

48224 

02237 

97703 

40 

4 

i:t 

16 

9.49677 

10.50423 

9.51819 

10.-18181 

10.02241 

9.97759 

45 

17 

4 

10 

49015 

50385 

51801 

48139 

02246 

97754 

44 

5 (> 

8 

17 

49054 

60340 

61903 

48097 

02250 

97750 

43 

52 

12 

18 

49092 

50308 

51940 

48054 

02264 

9774(1 

42 

48 

10 

19 

49730 

50270 

51988 

48012 

02258 

97742 

41 

44 

20 

20 

9.49708 

10.50232 

9.52031 

10.47909 

10.02202 

9.97738 

40 

40 

24 

21 

49806 

60194 

52073 

47927 

02200 

97734 

39 

36 

28 

22 

49844 

60150 

62115 

47885 

02271 

97720 

38 

32 

32 

23 

40382 

50118 

52157 

47843 

02275 

97725 

37 

28 

36 

24 

49920 

50080 

62200 

47800 

02279 

97721 

30 

24 

40 

26 

9.49958 

10.50042 

9.62242 

10.47758 

10.02283 

9.97717 

35 

20 

44 

20 

49996 

60004 

6228-4 

47710 

02287 

97713 

34 

10 

48 

27 

60031 

49900 

62326 

47071 

02292 

97708 

33 

12 

62 

28 

51K172 

49928 

52308 

47032 

02296 

97704 

32 

8 

60 

29 

60110 

49890 

62410 

47590 

02300 

97700 

31 

4 

14 

30 

9.50148 

10.49852 

9.62452 

10.47548 

10.02304 

9.97090 

30 

10 

4 

31 

50185 

49815 

62494 

47600 

02309 

97091 

29 

50 

8 

32 

50223 

49777 

52536 

47464 

02313 

97087 

28 

52 

12 

33 

60201 

49739 

62578 

47422 

02317 

97683 

27 

48 

10 

34 

50298 

49702 

52020 

47380 

02321 

97079 

20 

44 

20 

36 

9.50336 

10.49664 

9.52061 

10.47339 

10.02320 

9.97074 

25 

40 

24 

30 

60374 

49020 

62703 

47297 

02330 

97070 

24 

30 

28 

37 

50411 

49589 

52745 

47255 

02334 

97666 

23 

32 

32 

38 

60449 

49551 

62787 

47213 

02338 

97602 

22 

28 

30 

39 

50486 

49514 

52829 

47171 

02343 

97057 

21 

24 

40 

40 

9.50523 

10.49177 

9.52870 

10.47130 

'10.02347 

9.97658 

20 

20 

44 

41 

50501 

49439 

52912 

47088 

02351 

97649 

19 

10 

48 

42 

60598 

49402 

52953 

47017 

02356 

97645 

18 

12 

62 

43 

60035 

49305 

62995 

47005 

02360 

97640 

17 

8 

60 

44 

50073 

49327 

53037 

40903 

02364 

97636 

.16 

4 

15 

46 

9.50710 

10.49290 

9.53078 

10.40922 

10.02308 

9.97632 

15 

45 

4 

40 

50717 

49253 

63120 

40880 

02372 

97028 

14 

60 

8 

47 

60784 

49210 

53101 

. 40839 

02377 

97023 

13 

62 

12 

48 

50821 

49179 

53202 

40798 

02381 

97019 

12 

48 

10 

49 

50858 

49142 

53244 

46756 

02385 

97015 

11 

44 

20 

60 

9.'50896 

10.49104 

9.53285 

10.46715 

10.02390 

9.97010 

10 

40 

24 

61 

60933 

49067 

63327 

40073 

02391 

97006 

9 

30 

28 

62 

50970 

49030 

53308 

40632 

02398 

97002 

8 

32 

32 

63 

51007 

48993 

63409 

46591 

02403 

97597 

7 

28 

36 

54 

51013 

48957 

63450 

46550 

02407 

97693 

0 

24 

40 

65 

9.51080 

10.48920 

9.53492 

10.40508 

10.02411 

9.97589 

5 

20 

44 

50 

51117 

48883 

63533 

46467 

02416 

97584 

4 

10 

48 

57 

51154 

48840 

53574 

40420 

02420 

97580 

3 

12 

62 

68 

51191 

48809 

53615 

40385 

02424 

97576 

2 

8 

60 

59 

51227 

48773 

53056 

40344 

02429 

97571 

1 

4 

If. 

00 

61204 

48730 

53097 

40303 

02433 

97507 

0 

44 

M.S. 

7“ 

M 

10S' 

Cosine. 

D 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

71° 

M.S 

4 h 



















182 


Logarithms Trigonometric. 


l h 

19° 



Logarithms. 


160° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

16 

0 

9.51264 

10.48736 

9.53697 

10.46303 

10.02433 

9.97567 

60 

44 

4 

1 

51301 

48699 

53738 

46262 

02437 

97563 

59 

56 

8 

2 

51338 

48662 

53779 

46221 

02442 

97558 

58 

52 

12 

3 

51374 

48626 

53820 

46180 

02446 

97554 

57 

48 

16 

4 

51411 

48589 

53861 

46139 

02150 

97550 

56 

44 

20 

5 

9.51447 

10.48553 

9.53902 

10.46098 

10.02455 

9.97545 

55 

40 

24 

6 

51484 

48516 

53943 

46057 

02459 

97541 

54 

36 

28 

7 

51520 

48480 

53984 

46016 

02464 

97536 

53 

32 

32 

8 

51557 

48443 

54025 

45975 

• 02468 

97532 

52 

28 

36 

9 

51593 

48407 

54065 

45935 

02472 

97528 

51 

24 

40 

10 

9.51629 

10.48371 

9.54106 

10.45894 

10.02477 

9.97523 

50 

20 

44 

11 

51666 

48334 

54147 

45853 

02481 

97519 

49 

16 

48 

12 

51702 

48298 

54187 

45813 

02485 

97515 

48 

12 

52 

13 

51738 

48262 

54228 

45772 

02490 

97510 

47 

8 

56 

14 

51774 

48226 

54269 

45731 

02494 

97506 

46 

4 

IT 

15 

9.51811 

10.48189 

9.54309 

10.45691 

10.02499 

9.97501 

45 

43 

4 

16 

51847 

48153 

54350 

45650 

02503 

97497 

44 

56 

8 

17 

51883 

48117 

54390 

45610 

02508 

97492 

43 

52 

12 

18 

51919 

48081 

54431 

45569 

02512 

97488 

42 

48 

16 

19 

51955 

48045 

54471 

45529 

02516 

97484 

41 

44 

20 

20 

9.51991 

10.48009 

9.54512 

10.45488 

10.02521 

9.97479 

40 

40 

24 

21 

52027 

47973 

54552 

45448 

02525 

97475 

39 

36 

28 

22 

52063 

47937 

54593 

45407 

02530 

97470 

38 

32 

32 

23 

52099 

47901 

54633 

45367 

02534 

97466 

37 

28 

36 

24 

52135 

47865 

54673 

45327 

02539 

97461 

36 

24 

40 

25 

9.52171 

10.47829 

9.54714 

10.45286 

10.02543 

9.97457 

35 

20 

44 

26 

52207 

47793 

54754 

45246 

02547 

97453 

34 

16 

48 

27 

52242 

47758 

54794 

45206 

02552 

97448 

33 

12 

52 

28 

52278 

47722 

64835 

45165 

02556 

97444 

32 

8 

56 

29 

52314 

47686 

54S75 

45125 

02561 

97439 

31 

4 

18 

30 

9.52350 

10.47650 

9.54915 

, 10.45085 

10.02565 

9.97435 

30 

43 

4 

31 

52385 

47615 

54955 

45045 

02570 

97430 

29 

56 

8 

32 

52421 

47579 

54995 

45005 

02574 

97426 

28 

52 

12 

33 

52456 

47544 

55035 

44965 

02579 

97421 

27 

48 

16 

34 

52492 

47508 

55075 

44925 

02583 

97417 

26 

44 

20 

35 

9.52527 

10.47473 

9.55115 

10.44885 

10.02588 

9.97412 

25 

40 

24 

86 

52563 

47437 

55155 

44845 

02592 

97408 

24 

36 

28 

37 

52598 

47402 

55195 

44805 

02597 

97403 

23 

32 

32 

38 

52634 

47366 

55235 

44765 

02601 

97399 

22 

28 

36 

39 

52669 

47331 

55275 

44725 

02606 

97394 

21 

24 

40 

40 

9.52705 

10.47295 

9.55315 

10.44685 

10.02610 

9.97390 

20 

20 

44 

41 

52740 

47260 

55355 

44645 

02615 

97385 

19 

16 

48 

42 

52775 

47225 

55395 

44605 

02619 

97381 

18 

12 

52 

43 

52811 

47189 

55434 

44566 

02624 

97376 

17 

8 

56 

44 

52846 

47154 

55474 

44526 

02628 

97372 

16 

4 

19 

45 

9.52881 

10.47119 

9.55514 

10.44486 

10.02633 

9.97367 

15 

41 

4 

46 

52916 

47084 

55o54 

44446 

02637 

97363 

14 

56 

8 

47 

52951 

47049 

55593 

44407 

02642 

97358 

13 

52 

12 

48 

52986 

47014 

55633 

44367 

02647 

97353 

12 

48 

16 

49 

53021 

46979 

55673 

44327 

02651 

97349 

11 

44 

20 

50 

9.53056 

10.46944 

9.55712 

10.44288 

10.02656 

9.97344 

10 

40 

24 

51 

53092 

46908 

55752 

44248 

02660 

97340 

9 

36 

28 

52 

53126 

46874 

55791 

44209 

02665 

97335 

8 

32 

32 

53 

53161 

46839 

55831 

44169 

02669 

97331 

7 

28 

36 

54 

53196 

46804 

55870 

44130 

02674 

97326 

6 

24 

40 

55 

9.53231 

10.46769 

9.55910 

10.44090 

10.02678 

9.97322 

5 

20 

44 

56 

53266 

46734 

55949 

44051 

02683 

97317 

4 

16 

48 

57 

53301 

46699 

55989 

44011 

02688 

97312 

3 

12 

52 

58 

53336 

46664 

56028 

43972 

02692 

97308 

2 

8 

56 

59 

53370 

46630 

56067 

43933 

02697 

97303 

1 

4 

20 

60 

53405 

46595 

56107 

43893 

02701 

97299 

0 

40 

M.S. 

7 H 

M 

109 

Cosine. 

D 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

70° 

M.S. 

4 h 
















Logarithms Trigonometric. 185 


l b 

20° 



Logarithms. 


159° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

30 

0 

9.534)5 

10.46595 

9.56107 

10.43893 

10.02701 

9.97299 

60 

4:0 

4 

i 

53440 

46560 

56146 

43854 

02706 

97294 

59 

56 

8 

2 

53475 

46525 

60185 

43815 

02711 

97289 

58 

52 

12 

3 

53509 

46491 

56224 

43770 

02715 

97285 

57 

48 

16 

4 

53544 

46456 

56264 

43736 

02720 

97280 

56 

44 

20 

5 

9.53578 

10.46422 

9.56303 

10.43697 

10.02724 

9.97276 

55 

40 

24 

6 

53613 

46387 

56342 

43658 

02729 

97271 

54 

36 

28 

7 

53647 

46353 

56381 

43619 

02734 

97266 

53 

32 

32 

8 

53682 

46318 

56420 

43580 

02738 

97262 

52 

28 

36 

9 

53716 

46284 

56459 

43541 

02743 

97257 

51 

24 

40 

10 

9.53751 

10.46249 

9.56498 

10.43502 

10.02748 

9.97252 

50 

20 

44 

11 

53785 

46215 

56537 

• 43463 

02752 

97248 

49 

16 

48 

12 

53819 

46181 

56576 

43424 

02757 

97243 

48 

12 

52 

13 

53854 

46146 

56615 

43385 

02762 

97238 

47 

8 

56 

14 

53888 

46112 

56654 

43346 

02766 

97234 

46 

4 

31 

15 

9.53922 

10.46078 

9.56693 

10.43307 

10.02771 

9.97229 

45 

3<J 

4 

16 

53957 

46043 

56732 

43268 

02776 

97224 

44 

56 

8 

17 

63991 

46009 

56771 

43229 

02780 

97220 

43 

52 

12 

18 

54025 

45975 

56810 

43190 

02785 

97215 

42 

48 

16 

19 

54059 

45941 

56849 

43151 

02790 

97210 

41 

44 

20 

20 

9.54093 

10.45907 

9.56887 

10.43113 

10.02794 

9.97206 

40 

40 

24 

21 

54127 

45873 

56926 

43074 

02799 

97201 

39 

36 

28 

22 

54161 

45839 

56965 

43035 

02804 

97196 

38 

32 

32 

23 

54195 

45805 

57004 

42996 

02808 

97192 

37 

28 

36 

24 

54229 

45771 

57042 

42958 

02813 

97187 

36 

24 

40 

25 

9.54263 

10.45737 

9.57081 

10.42919 

10.02818 

9.97182 

35 

20 

44 

26 

54297 

45703 

57120 

42880 

02822 

97178 

34 

16 

48 

27 

54331 

45669 

67158 

42842 

02827 

97173 

33 

12 

52 

28 

54365 

45635 

57197 

42803 

02832 

97168 

32 

8 

56 

29 

54399 

45601 

57235 

42765 

02837 

97163 

31 

4 

33 

30 

9.54433 

10.45567 

9.57274 

10.42726 

10.02841 

9.97159 

30 

38 

4 

31 

54466 

45534 

57312 

42688 

02846 

97154 

29 

56 

8 

32 

54500 

45500 

57351 

42649 

02851 

97149 

28 

52 

12 

33 

54534 

45466 

57389 

42611 

02855 

97145 

27 

48 

16 

34 

54567 

45433 

57428 

42572 

02860 

97140 

26 

44 

20 

35 

9.54601 

10.45399 

9.57466 

10.42534 

10.02865 

9.97135 

25 

40 

24 

36 

54635 

45365 

57504 

42496 

02870 

97130 

24 

36 

28 

37 

54668 

45332 

57543 

42457 

02874 

97126 

23 

32 

32 

38 

54702 

45298 

57581 

42419 

02879 

97121 

22 

28 

36 

39 

54735 

45265 

57619 

42381 

02884 

97116 

21 

24 

40 

40 

9.54769 

10.45231 

9.57658 

10.42342 

1C.02889 

9.97111 

20 

20 

44 

41 

54802 

45198 

57696 

42304 

02893 

97107 

19 

16 

48 

42 

54836 

45164 

57734 

42266 

02898 

97102 

18 

12 

52 

43 

54869 

45131 

57772 

42228 

02903 

97097 

17 

8 

56 

44 

64903 

45097 

57810 

42190 

02908 

97092 

16 

4 

33 

45 

9.54936 

10.45064 

9.57849 

10.42151 

10.02913 

9.97087 

15 

37 

4 

46 

54969 

45031 

57887 

42113 

02917 

97083 

14 

56 

8 

47 

55003 

44997 

57925 

42075 

02922 

97078 

13 

52 

12 

48 

55036 

44964 

57963 

42037 

02927 

97073 

12 

48 

16 

49 

55069 

44931 

58001 

41999 

02932 

97068 

11 

44 

20 

50 

9.55102 

10.44898 

9.58039 

10.41961 

10.02937 

9.97063 

10 

40 

24 

51 

65136 

44864 

58077 

41923 

02941 

97059 

9 

36 

28 

52 

55169 

44831 

58115 

41885 

02946 

97054 

8 

32 

32 

53 

55202 

44798 

58153 

41847 

02951 

97049 

7 

28 

36 

54 

55235 

44765 

58191 

41809 

02956 

97044 

6 

24 

40 

55 

9.55268 

10.44732 

9.58229 

10.41771 

10.02961 

9.97039 

5 

20 

44 

56 

55301 

44699 

58267 

41733 

02965 

97035 

4 

16 

4« 

57 

55334 

44666 

58304 

41696 

02970 

97030 

3 

12 

52 

58 

55367 

44633 

58342 

41658 

02975 

97025 

2 

8 

56 

59 

55400 

44600 

58380 

41620 

02980 

97020 

1 

4 

24: 

60 

55433 

44567 

58418 

41582 

02985 

97015 

0 

36 

M. S. 

7 h 

M 

110 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

69° 

M. S. 

4 U 


















184 Logarithms Trigonometric. 


l b 

21 c 



Logarithms. 


158° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

34 

0 

9.55433 

10.44567 

9.58418 

10.41582 

10.02985 

9.97015 

60 

36 

4 

1 

55466 

44534 

58455 

41545 

02990 

97010 

59 

56 

8 

2 

55499 

44501 

58493 

' 41507 

02995 

97005 

58 

52 

12 

3 

55532 

44468 

58531 

41469 

02999 

97001 

57 

48 

16 

4 

55564 

44436 

58569 

41431 

03004 

96996 

56 

44 

20 

5 

9.55597 

10.44403 

9.58606 

10.41394 

10.03009 

9.96991 

55 

40 

24 

6 

55630 

44370 

58644 

41356 

03014 

969S6 

54 

36 

28 

.7 

55663 

44337 

58681 

41319 

03019 

96981 

53 

32 

32 

8 

55695 

44305 

58719 

41281 

03024 

96976 

52 

28 

36 

9 

55728 

44272 

58757 

41243 

03029 

96971 

51 

24 

40 

10 

9.55761 

10.44239 

9.58794 

10.41206 

10.03034 

9.96966 

50 

20 

44 

11 

55793 

44207 

58832 , 

41168 

03038 

96962 

49 

16 

48 

12 

55826 

44174 

58869 

41131 

03043 

96957 

48 

12 

62 

13 

55858 

44142 

58907 

41093 

03048 

96952 

47 

8 

66 

14 

55891 

44109 

58944 

41056 

03053 

96947 

46 

4 

35 

15 

9.55923 

10.44077 

9.58981 

10.41019 

10.03058 

9.96942 

45 

35 

4 

16 

55956 

44044 

59019 

40981 

03063 

96937 

44 

56 

8 

17 

55988 

44012 

59056 

40944 

03068 

96932 

43 

52 

12 

18 

56021 

43979 

59094 

40906 

03073 

96927 

42 

48 

16 

19 

56053 

43947 

59131 

40869 

03078 

96922 

41 

44 

20 

20 

9.56085 

10.43915 

9.59168 

10.40832 

10.03083 

9.96917 

40 

40 

24 

21 

56118 

43882 

592U5 

40795 

03088 

96912 

39 

36 

28 

22 

56150 

43850 

59243 

40757 

03093 

96907 

38 

32 

32 

23 

56182 

43818 

59280 

40720 

03097 

96903 

37 

28 

36 

24 

56215 

43785 

59317 

40683 

03102 

96S98 

36 

24 

40 

25 

9.56247 

10.43753 

9.59354 

10.40646 

10.03107 

9.96893 

35 

20 

44 

26 

56279 

43721 

59391 

40609 

03112 

96888 

34 

16 

48 

27 

56311 

43689 

59429 

40571 

03117 

96883 

33 

12 

52 

28 

56343 

43657 

59466 

40534 

03122 

96878 

32 

8 

56 

29 

56375 

43625 

59503 

40497 

03127 

96873 

31 

4 

3G 

30 

9.56408 

10.43592 

9.59540 

10.40460 

10.03132 

9.96868 

30 

34 

4 

31 

56440 

43560 

59577 

40423 

03137 

96863 

29 

56 

8 

32 

56472 

43528 

59614 

40386 

03142 

96S58 

28 

52 

12 

33 

56504 

43496 

59651 

40349 

03147 

96853 

27 

48 

16 

34 

56536 

43464 

59688 

40312 

03152 

96848 

26 

44 

20 

35 

9.56568 

10.43432 

9.59725 

10.40275 

10.03157 

9.96843 

25 

40 

24 

36 

56599 

43401 

59762 

40238 

03162 

96838 

24 

36 

28 

37 

56631 

43369 

59799 

40201 

03167 

96833 

23 

32 

32 

38 

56663 

43337 

59835 

40165 

03172 

96828 

22 

28 

36 

39 

56695 

43305 

59872 

40128 

03177 

96823 

21 

24 

40 

40 

9.56727 

10.43273 

9.59909 

10.40091 

10.03182 

9.96818 

20 

20 

44 

41 

56759 

43241 

59946 

40054 

03187 

96813 

19 

16 

48 

42 

56790 

43210 

59983 

40017 

03192 

96808 

18 

12 

52 

43 

56822 

43178 

60019 

39981 

03197 

• 96803 

17 

8 

5G 

44 

56854 

43146 

60056 

39944 

03202 

96798 

16 

4 

27 

45 

9.56886 

10.43114 

9.60093 

10.39907 

10.03207 

9.96793 

15 

33 

4 

46 

66917 

43083 

60130 

39870 

03212 

96788 

14 

56 

8 

47 

56949 

43051 

60166 

39834 

03217 

96783 

13 

52 

12 

48 

56980 

43020 

60203 

39797 

03222 

96778 

12 

48 

16 

49 

57012 

42988 

60240 

39760 

03228 

96772 

11 

44 

20 

50 

9.57044 

10.42956 

9.60276 

10.39724 

10.03233 

9.96767 

10 

40 

24 

51 

57075 

42925 

60313 

39687 

03238 

96762 

9 

36 

28 

52 

57107 

42893 

60349 

39651 

03243 

96757 

8 

32 

32 

53 

57138 

42862 

60386 

39614 

03248 

96752 

7 

28 

36 

54 

57169 

42831 

60422 

39578 

03253 

96747 

6 

24 

40 

.55 

9.57201 

10.42799 

9.60459 

10.39541 

10.03258 

9.96742 

5 

20 

44 

56 

57232 

42768 

60495 

39505 

03263 

96737 

4 

16 

48 

57 

57264 

42736 

60532 

39468 

03268 

96732 

3 

12 

52 

58 

57295 

42705 

60568 

39432 

03273 

96727 

2 

8 

56 

59 

57326 

42674 

60605 

39395 

03278 

96722 

1 

4 

38 

60 

57358 

42642 

60641 

39359 

03283 

96717 

0 

32 

M.S. 

7 h 

M 

111 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

68 ° 

M. S. 

4 h 

















Logarithms Trigonometric. 186 


l h 

22 ° 



Logarithms. 


157° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

38 

0 

9.57358 

10.42642 

9.60641 

10.39359 

10.03283 

9.96717 

60 

3 3 

4 

1 

57389 

42611 

60677 

39323 

03289 

96711 

59 

56 

8 

2 

57420 

42580 

60714 

39286 

03294 

96706 

58 

52 

12 

3 

57451 

42549 

60750 

39250 

03299 

96701 

57 

48 

16 

4 

57482 

42518 

60786 

39214 

03304 

96696 

56 

44 

20 

5 

9.57514 

10.42486 

9.60823 

10.39177 

10.03309 

9.96691 

55 

40 

24 

6 

57545 

42455 

60859 

39141 

03314 

96686 

54 

36 

28 

7 

57576 

42424 

60895 

39105 

03319 

96681 

53 

32 

32 

8 

57607 

42393 

60931 

39069 

03324 

96676 

52 

28 

36 

9 

57638 

42362 

60967 

39033 

03330 

96670 

51 

24 

40 

10 

9.57669 

10.42331 

9.61004 

10.38996 

10.03335 

9.96665 

50 

20 

44 

11 

57700 

42300 

61040 

38960 

03340 

96660 

49 

16 

48 

12 

57731 

42269 

61076 

38924 

03345 

96655 

48 

12 

52 

13 

57762 

42238 

61112 

38888 

03350 

96650 

47 

8 

56 

14 

57793 

42207 

61148 

38852 

03355 

96615 

46 

4 

39 

15 

9.57824 

10.42170 

9.61184 

10.38816 

10.03360 

9.96640 

45 

31 

4 

16 

57855 

42145 

61220 

38780 

03366 

96634 

44 

56 

8 

17 

57885 

42115 

61256 

38744 

03371 

96629 

43 

52 

12 

18 

57916 

42084 

61292 

3S708 

03376 

96624 

42 

48 

16 

19 

57947 

42053 

61328 

38672 

08381 

96619 

41 

44 

20 

20 

9.57978 

10.42022 

9.61364 

10.38636 

10.03386 

9.96614 

40 

40 

24 

21 

58008 

41992 

61400 

38600 

03392 

96608 

39 

36 

28 

22 

58039 

41961 

61436 

38564 

03397 

96603 

38 

32 

32 

23 

58070 

41930 

61472 

38528 

03402 

96598 

37 

28 

36 

24 

58101 

41899 

61508 

38492 

09407 

96593 

36 

24 

40 

25 

9.58131 

10.41869 

9.61544 

10.38456 

10.03412 

9.96588 

35 

20 

44 

26 

58162 

41838 

61579 

38421 

03418 

96582 

34 

16 

48 

27 

58192 

41808 

61615 

38385 

03423 

96577 

33 

12 

52 

28 

58223 

41777 

61651 

38349 

03428 

96572 

32 

8 

56 

29 

58253 

41747 

61687 

38313 

03433 

96567 

31 

4 

30 

30 

9.58284 

10.41716 

9.61722 

10.38278 

10.03438 

9.96562 

30 

30 

4 

31 

58314 

41686 

■61758 

38242 

03444 

96556 

29 

56 

8 

32 

58345 

41655 

61794 

38206 

03449 

96551 

28 

52 

12 

33 

58375 

41625 

61830 

38170 

03454 

96546 

27 

48 

16 

34 

58406 

41594 

61865 

38135 

03459 

96541 

26 

44 

20 

35 

9.58436 

10.41564 

9.61901 

10.38099 

10.03465 

9.96535 

25 

40 

24 

36 

58467 

41533 

61936 

38064 

03470 

96530 

24 

36 

28 

37 

58497 

41503 

61972 

38028 

03475 

96525 

23 

32 

32 

38 

58527 

41473 

62008 

37992 

03480 

96520 

22 

28 

36 

39 

58557 

41443 

62043 

37957 

03486 

96514 

21 

24 

40 

40 

9.58588 

10.41412 

9.62079 

10.37921 

10.03491 

9.96509 

20 

20 

44 

•41 

58618 

41382 

62114 

37886 

03496 

96504 

19 

16 

48 

42 

58648 

41352 

62150 

37850 

03502 

96498 

18 

12 

52 

43 

58678 

41322 

62185 

37815 

03507 

96493 

17 

8 

56 

44 

58709 

41291 

62221 

37779 

03512 

96488 

16 

4 

31 

45 

9.58739 

10.41261 

9.62256 

10.37744 

10.03517 

9.96483 

15 

30 

4 

46 

58769 

41231 

62292 

37708 

03523 

96477 

14 

56 

8 

47 

58799 

41201 

62327 

37673 

03528 

96472 

13 

52 

12 

48 

58829 

41171 

62362 

37638 

03533 

96467 

12 

48 

16 

49 

58859 

41141 

62398 

37602 

03539 

96461 

11 

41 

20 

50 

9.58889 

10.41111 

9.62433 

10.37567 

10.03544 

9.96456 

10 

40 

24 

51 

58919 

41081 

62468 

37532 

03549 

96451 

9 

36 

28 

52 

58949 

41051 

62504 

37496 

03555 

96415 

8 

32 

32 

53 

58979 

41021 

62539 

37461 

03560 

96440 

7 

28 

36 

54 

59009 

40991 

62574 

37426 

03565 

96435 

6 

24 

40 

55 

9.59039 

10.40961 

9.62609 

10.37391 

10.03571 

9.96429 

5 

20 

44 

56 

59069 

40931 

62645 

37355 

03576 

9(5424 

4 

16 

48 

57 

59098 

40902 

62680 

37320 

03581 

96419 

3 

12 

52 

58 

59128 

40872 

62715 

37285 

03587 

96413 

2 

8 

56 

59 

59158 

40842 

62750 

37250 

03592 

96408 

1 

4 

33 

60 

59188 

40812 

62785 

37215 

03597 

9(5403 

0 

38 

M.S. 

7** 

M 

112 c 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

67° 

M.S. 

ill 























186 


Logarithms Trigonometric. 


l h 

23° 



Logarithms. 


156° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

1 M 

M.S. 

32 

0 

9.59188 

10.40812 

9.62785 

10.37215 

10.03597 

9.96403 

60 

as 

4 

1 

59218 

40782 

62820 

37180 

03603 

96397 

59 

56 

8 

2 

59247 

40753 

62855 

37145 

03608 

96392 

58 

52 

12 

3 

59277 

40723 

62890 

37110 

03613 

96387 

57 

48 

16 

4 

59307 

40693 

62926 

37074 

03619 

96381 

56 

44 

20 

5 

9.59336 

10.40664 

9.62961 

10.37039 

10.03624 

9.96376 

55 

40 

24 

6 

59366 

40634 

62996 

37004 

03630 

96370 

54 

36 

28 

7 

59396 

40604 

63031 

36969 

03635 

96365 

53 

32 

32 

8 

59425 

40575 

63066 

36934 

03640 

96360 

52 

28 

36 

9 

59455 

40545 

63101 

36899 

03646 

96354 

51 

24 

40 

10 

9.59484 

10.40516 

9.63135 

10.36865 

10.03651 

9.96349 

50 

20 

44 

11 

59514 

40486 

63170 

36830 

03657 

96343 

49 

16 

48 

12 

59543 

40457 

63205 

36795 

03662 

96338 

48 

12 

52 

13 

59573 

40427 

63240 

36760 

03667 

96333 

47 

8 

56 

14 

59602 

40398 

63275 

36725 

03673 

96327 

46 

4 

33 

15 

9.59632 

10.40368 

9.63310 

10.36690 

10.03678 

9.96322 

45 

27 

4 

16 

59661 

40339 

63345 

36655 

03684 

96316 

44 

56 

8 

17 

59690 

40310 

63379 

36621 

03689 

96311 

43 

52 

12 

18 

59720 

40280 

63414 

36586 

03695 

96305 

42 

48 

16 

19 

59749 

40251 

63449 

.36551 

03700 

96300 

41 

44 

20 

20 

9.59778 

10.40222 

9.63484 

10.36516 

10.03706 

9.96294 

40 

40 

24 

21 

59S08 

40192 

63519 

30481 

03711 

96289 

39 

36 

28 

22 

59837 

40163 

63553 

36447 

03716 

96284 

38 

32 

32 

23 

59866 

40134 

63588 

36412 

03722 

96278 

37 

28 

36 

24 

59895 

40105 

63623 

36377 

09727 

96273 

36 

24 

40 

25 

9.59924 

10.40076 

9.63657 

1,6.36343 

10.03733 

9.96267 

35 

20 

44 

26 

59954 

40016 

63692 

36308 

03738 

96262 

34 

16 

48 

27 

59983 

40017 

63726 

36274 

03744 

96256 

33 

12 

52 

28 

60012 

39988 

63761 

36239 

03749 

96251 

32 

8 

56 

29 

60011 

39959 

63796 

36204 

03755 

96245 

31 

4 

3L 

30 

9.60070 

10.39930 

9.63830 

. 10.36170 

10.03760 

9.96240 

30 

26 

4 

31 

60099 

39901 

63865 

36135 

03766 

96234 

29 

56 

8 

32 

60128 

39872 

63899 

36101 

03771 

96229 

28 

52 

12 

33 

60157 

39843 

63934 

36066 

03777 

96223 

27 

48 

16 

34 

60186 

39814 

63968 

36032 

03782 

96218 

26 

41 

20 

35 

9.60215 

10.39785 

9.64003 

10.35997 

10.03788 

9.96212 

25 

40 

24 

36 

60244 

39756 

64037 

35963 

03793 

96207 

24 

36 

28 

37 

60273 

39727 

64072 

35928 

03799 

96201 

23 

32 

32 

38 

60302 

39698 

64100 

35894 

03804 

96196 

22 

28 

36 

39 

60331 

39669 

64140 

35860 

03810 

96190 

21 

24 

40 

40 

9.60359 

10.39641 

9.64175 

10.35825 

10.03815 

9.96185 

20 

20 

44 

41 

60388 

39612 

64209 

35791 

03821 

96179 

19 

16 

48 

42 

60417 

39583 

64243 

35757 

03826 

96174 

18 

12 

52 

43 

60446 

39554 

64278 

35722 

03832 

96168 

17 

8 

56 

44 

60474 

39526 

64312 

35688 

03838 

96162 

16 

4 

33 

45 

9.60503 

10.39497 

9.64346 

10.35654 

10.03843 

9.96157 

15 

25 

4 

46 

60532 

39468 

64381 

35619 

03849 

96151 

14 

56 

8 

47 

60561 

39439 

64415 

35585 

03854 

96146 

13 

52 

12 

48 

60589 

39411 

64449 

35551 

03860 

96140 

12 

48 

16 

49 

60618 

39382 

64483 

35517 

03865 

96135 

11 

44 

2C 

50 

9.60646 

10.39354 

9.64517 

10.35483 

10.03871 

9.96129 

10 

40 

24 

61 

60675 

39325 

64552 

35448 

03877 

96123 

9 

36 

28 

52 

60704 

39296 

64586 

35414 

03882 

96118 

8 

32 

32 

53 

60732 

39268 

64620 

35380 

03888 

96112 

7 

28 

36 

54 

60761 

39239 

64654 

35346 

03893 

96107 

6 

24 

40 

55 

9.60789 

10.39211 

9.64688 

10.35312 

10.03899 

9.96101 

5 

20 

44 

66 

60818 

39182 

64722 

35278 

03905 

96095 

4 

16 

48 

57 

60846 

39154 

64756 

35244 

03910 

96090 

3 

12 

52 

58 

60875 

39125 

64790 

35210 

03916 

96084 

2 

8 

56 

59 

60903 

39097 

64824 

35176 

03921 

96079 

1 

4 

3G 

60 

60931 

39069 

64858 

35142 

03927 

96073 

0 

24: 

M.S. 

?h 

M 

113 c 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

66° 

M. S. 

4 U 




















Logarithms Trigonometric. 


187 


l b 

24° 



Logarithms. 


155° 

10 h 

M.S. 

M 

Siue. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

36 

0 

9.60931 

10.39069 

9.64858 

10.35142 

10.03927 

9.96073 

60 

24: 

4 

1 

60960 

39040 

64892 

35108 

03933 

96067 

59 

56 

8 

2 

60988 

39012 

64926 

35074 

03938 

96062 

58 

52 

12 

3 

61016 

38984 

64960 

35040 

03944 

96056 

57 

48 

16 

4 

61045 

38955 

64994 

35006 

03950 

96050 

56 

44 

20 

5 

9.61073 

10.38927 

9.65028 

10.34972 

10.03955 

9.96045 

55 

40 

24 

6 

61101 

38899 

65062 

34938 

03961 

96039 

54 

36 

28 

7 

61129 

38871 

65096 

34904 

03966 

96034 

53 

32 

32 

8 

61158 

38842 

65130 

' 34870 

03972 

96028 

52 

28 

36 

9 

61186 

38814 

65164 

34836 

03978 

96022 

51 

24 

40 

10 

9.61214 

10.38786 

9.65197 

10.34803 

10.03983 

9.96017 

50 

20 

44 

11 

61242 

38758 

6523 L 

34769 

03989 

96011 

49 

16 

48 

12 

61270 

38730 

65265 

34735 

03995 

96005 

48 

12 

52 

13 

61298 

38702 

65299 

34701 

04000 

96000 

47 

8 

56 

14 

61326 

38674 

65333 

34667 

04006 

95994 

46 

4 

37 

15 

9.61354 

10.38646 

9.65366 

10.34634 

10.04012 

9.95988 

45 

23 

4 

16 

61382 

38618 

65400 

34600 

04018 

95982 

44 

56 

8 

17 

61411 

38589 

65434 

34566 

04023 

95977 

43 

52 

12 

18 

61438 

38562 

65467 

34533 

04029 

95971 

42 

48 

16 

19 

61466 

38534 

65501 

34499 

04035 

95965 

41 

44 

20 

20 

9.61494 

10.38506 

9.65535 

10.34465 

10.04040 

9.95960 

40 

40 

24 

21 

61522 

38478 

65568 

34432 

04046 

95954 

39 

36 

28 

22 

61550 

38450 

65602 

34398 

04052 

95948 

38 

32 

32 

23 

61578 

38422 

65636 

34364 

04058 

95942 

37 

28 

36 

24 

61606 

38394 

65669 

34331 

04063 

95937 

36 

24 

40 

25 

9.61634 

10.38366 

9.65703 

10.34297 

10.04069 

9.95931 

35 

20 

44 

26 

61662 

38338 

65736 

34264 

04075 

95925 

34 

16 

48 

27 

61689 

38311 

65770 

34230 

04080 

95920 

33 

12 

52 

28 

61717 

38283 

65803 

34197 

04086 

95914 

32 

8 

56 

29 

61745 

38255 

65837 

34163 

04092 

95908 

31 

4 

38 

30 

9.61773 

10.38227 

9.65870 

10.34130 

10.04098 

9.95902 

30 

22 

4 

31 

61800 

38200 

65904 

34096 

04103 

95897 

29 

56 

8 

32 

61828 

38172 

65937 

34063 

04109 

95891 

28 

52 

12 

33 

61856 

38144 

65971 

34029 

04115 

95885 

27 

48 

16 

34 

61883 

38117 

66004 

33996 

04121 

95879 

26 

44 

20 

35 

9.61911 

10.38089 

9.66038 

10.33962 

10.04127 

9.95873 

25 

40 

24 

36 

61939 

38061 

66071 

33929 

04132 

95868 

24 

36 

28 

37 

61966 

38034 

66104 

33896 

04138 

95862 

23 

32 

32 

38 

61994 

38006 

66138 

33862 

04144 

95856 

22 

28 

36 

39 

6202 L 

37979 

66171 

33829 

04150 

95850 

21 

24 

40 

40 

9.62049 

10.37951 

9.66204 

10.33796 

10.04156 

9.95844 

20 

20 

44 

41 

62076 

37924 

66238 

33762 

04161 

95839 

19 

16 

48 

42 

62104 

37896 

66271 

33729 

04167 

95833 

18 

12 

52 

43 

62131 

37869 

66304 

33696 

04173 

95827 

17 

8 

56 

44 

62159 

37841 

66337 

33663 

04179 

95821 

16 

4 

39 

45 

9.62186 

10.37814 

9.66371 

10.33629 

10.04185 

9.95815 

15 

21 

4 

46 

62214 

37786 

66404 

33596 

04190 

95810 

14 

56 

8 

47 

62241 

37759 

66437 

33563 

04196 

95804 

13 

52 

12 

48 

62268 

37732 

66470 

33530 

04202 

95798 

12 

48 

16 

49 

62296 

37704 

66503 

33497 

04208 

95792 

11 

44 

20 

50 

9.62323 

10.37677 

9.66537 

10.33463 

10.04214 

9.95786 

10 

40 

24 

51 

62350 

37650 

66570 

33430 

04220 

95780 

9 

36 

28 

52 

62377 

37623 

66603 

33397 

04225 

95775 

8 

32 

32 

53 

62405 

37595 

66636 

33364 

04231 

95769 

7 

28 

36 

54 

62432 

37568 

66669 

33331 

04237 

95763 

6 

24 

40 

55 

9.62459 

10.37541 

9.66702 

10.33298 

10.04243 

9.95757 

5 

20 

44 

56 

62486 

37514 

66735 

33265 

04249 

95751 

4 

16 

48 

57 

62513 

37487 

66768 

33232 

04255 

95745 

3 

12 

52 

58 

62541 

37459 

66801 

33199 

04261 

95739 

2 

8 

56 

59 

62568 

37432 

66834 

33166 

04267 

95733 

1 

4 

40 

60 

62595 

37405 

66867 

33133 

04272 

95728 

0 

20 

M. S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M. S. 

7 b 

114 

0 






65°) 

4 h 
























183 Logarithms Trigonometric. 


l h 

25 c 



IiOgaritlims. 


154 c 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

| Cosine. 

M 

M. S. 

40 

0 

9.62595 

10.37405 

9.66867 

10.33133 

10.04272 

9.95728 

60 

30 

4 

1 

62622 

37378 

66900 

33100 

04278 

95722 

59 

56 

8 

2 

62649 

37351 

66933 

33067 

04284 

95716 

58 

52 

12 

3 

62676 

37324 

66966 

33034 

04290 

95710 

57 

48 

16 

4 

. 62703 

37297 

66999 

33001 

04296 

95704 

56 

44 

20 

5 

9.62730 

10.37270 

9.67032 

10.32968 

10.04302 

9.95698 

55 

40 

24 

6 

62757 

37243 

67065 

33935 

04308 

95692 

54 

36 

28 

7 

62784 

37216 

67098 

32902 

04314 

95686 

53 

32 

32 

8 

62811 

37189 

67131 

32869 

04320 

95680 

52 

28 

36 

9 

6283S 

37162 

67163 

32837 

04326 

95674 

51 

24 

40 

10 

9.62865 

10.37135 

9.67196 

10.32804 

10.04332 

9.95668 

50 

20 

44 

11 

62892 

37108 

67229 

32771 

04337 

95663 

49 

16 

48 

12 

62918 

37082 

67262 

32738 

04343 

95657 

48 

12 

52 

13 

62945 

37055 

67295 

32705 

04349 

95651 

47 

8 

56 

14 

62972 

37028 

67327 

32673 

04355 

95645 

46 

4 

41 

15 

9.62999 

10.37001 

9.67360 

10.32640 

10.04361 

9.95639 

45 

19 

4 

16 

63026 

36974 

67393 

32607 

04367 

95633 

44 

56 

8 

17 

63052 

36948 

67426 

32574 

04373 

95627 

43 

52 

12 

18 

63079 

36921 

67458 

32542 

04379 

95621 

42 

48 

16 

19 

63106 

36894 

67491 

32509 

04385 

95615 

41 

44 

20 

20 

9.63133 

10.36867* 

9.67524 

10.32476 

10.04391 

9.95609 

40 

40 

24 

21 

63159 

36841 

67556 

32444 

04397 

95603 

39 

36 

28 

22 

63186 

36814 

67589 

32411 

04403 

95597 

38 

32 

32 

23 

63213 

36787 

67622 

32378 

04409 

95591 

37 

28 

36 

24 

63239 

36761 

67654 

32346 

04415 

95585 

36 

24 

40 

25 

9.63266 

10.36734 

9.67687 

10.32313 

10.04421 

9.95579 

35 

20 

44 

26 

63292 

36708 

67719 

32281 

04427 

95573 

34 

16 

48 

27 

63319 

36681 

67752 

32248 

04433 

95567 

33 

12 

52 

28 

63345 

36655 

67785 

32215 

04439 

95561 

32 

8 

56 

29 

63372 

36628 

67817 

32183 

04445 

95555 

31 

4 

43 

30 

9.63398 

10.36602 

9.67850 

10.32150 

10.04451 

9.95549 

30 

18 

4 

31 

63425 

36575 

67882 

32118 

04457 

95543 

29 

56 

8 

32 

63451 

36549 

67915 

32085 

04463 

95537 

28 

52 

12 

33 

63478 

36522 

67947 

32053 

04469 

95531 

27 

43 

16 

34 

63504 

36496 

67980 

32020 

04475 

95525 

26 

44 

20 

35 

9.63531 

10.36469 

9.68012 

10.31988 

10.04481 

9.95519 

25 

40 

24 

36 

63557 

36443 

68044 

31956 

04487 

95513 

24 

36 

28 

37 

63583 

36417 

68077 

31923 

04493 

95507 

23 

32 

32 

38 

63610 

36390 

68109 

31891 

04500 

95500 

22 

28 

36 

39 

63636 

36364 

68142 

31858 

04506 

95494 

21 

24 

40 

40 

9.63662 

10.36338 

9.68174 

10.31826 

10.04512 

9.95488 

20 

20 

44 

41 

63689 

36311 

68206 

31794 

04518 

95482 

19 

16 

48 

42 

63715 

36285 

68239 

31761 

04524 

95476 

18 

12 

52 

43 

63741 

36259 

68271 

31729 

04530 

95470 

17 

8 

56 

44: 

63767 

36233 

68303 

31697 

04536 

95464 

16 

4 

43 

45 

9.63794 

10.36206 

9.Q8336 

10.31664 

10.04542 

9.95458 

15 

17 

4 

46 

63820 

36180 

68368 

31632 

04548 

95452 

14 

56 

8 

47 

63846 

36154 

68400 

31600 

04554 

95446 

13 

52 

12 

48 

63872 

36128 

68432 

31568 

04560 

95440 

12 

48 

16 

49 

63898 

36102 

68465 

31535 

04566 

95434 

11 

44 

20 

50 

9.63924 

10.36076 

9.68497 

10.31503 

10.04573 

9.95427 

10 

40 

24 

51 

63950 

36050 

68529 

31471 

04579 

95421 

9 

36 

28 

52 

63976 

36024 

68561 

31439 

04585 

95415 

8 

32 

32 

53 

64002 

35998 

68593 

31407 

04591 

95409 

7 

28 

36 

54 

64028 

35972 

68626 

31374 

04597 

95403 

6 

24 

40 

55 

9.64054 

10.35946 

9.68658 

10.31342 

10.04603 

9.95397 

5 

20 

44 

56 

64080 

35920 

68690 

31310 

04609 

95391 

4 

16 

48 

57 

64106 

35894 

68722 

31278 

04616 

95384 

3 

12 

52 

58 

64132 

35.868 

68754 

31246 

04622 

95378 

2 

8 

56 

59 

64158 

35842 

68786 

31214 

04628 

95372 

1 

4 

44 

60 

64184 

35816 

68818 

31182 

04634 

95366 

0 

16 

M. S. 

7 h | 

M 

115 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine, 

M 

64° 

> 1 . s. 

4 h 



















Logarithms Trigonometric. 


l h 

26° 



Logarithms. 


153° 

10 h 

M.S 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

44L 

0 

9.64184 

10.35816 

9.68818 

10.31182 

10.04634 

9.95366 

60 

lf> 

4 

1 

64210 

35790 

68850 

31150 

04640 

95360 

59 

56 

8 

2 

64236 

35764 

68882 

31118 

04646 

95354 

58 

52 

12 

3 

64262 

35738 

68914 

31086 

04652 

95348 

57 

48 

16 

4 

64288 

35712 

68946 

31054 

04659 

95341 

56 

44 

20 

5 

9.64313 

10.35687 

9.68978 

10.31022 

10.04665 

9.95335 

55 

40 

24 

6 

64339 

35661 

69010 

30990 

04671 

95329 

54 

36 

28 

7 

64365 

35635 

' 69042 

30958 

04677 

95323 

53 

32 

32 

8 

64391 

35609 

69074 

30926 

04683 

95317 

52 

28 

30 

9 

64417 

35583 

69106 

30894 

04690 

95310 

5L 

24 

40 

10 

9.64442 

10.35558 

9.69138 

10.30862 

10.04696 

9.95304 

50 

20 

44 

11 

64468 

35532 

69170 

30830 

04702 

95298 

49 

16 

48 

12 

64494 

35506 

69202 

30798 

04708 

95292 

48 

12 

52 

13 

64519 

35481 

69234 

30766 

04714 

95286 

47 

8 

56 

14 

64545 

35455 

69266 

30734 

04721 

95279 

46 

4 

45 

15 

9.64571 

10.35429 

9.69298 

10.30702 

10.04727 

9.95273 

45 

15 

4 

16 

64596 

35404 

69329 

30671 

04733 

95267 

44 

56 

8 

17 

64622 

35378 

69361 

30639 

04739 

95261 

43 

52 

12 

18 

64647 

35353 

69393 

30607 

04746 

95254 

42 

48 

16 

19 

64673 

35327 

69425 

30575 

04752 

95248 

41 

44 

20 

20 

9.64698 

10.35302 

9.69457 

10.30543 

10.04758 

9.95242 

40 

40 

24 

21 

64724 

35276 

69 488 

30512 

04764 

95236 

39 

36 

28 

22 

64749 

35251 

69520 

30480 

04771 

95229 

38 

32 

32 

23 

64775 

35225 

69552 

30448 

04777 

95223 

37 

28 

36 

24 

64800 

35200 

69584 

30416 

04783 

95217 

36 

24 

40 

25 

9.64826 

10.35174 

9.69615 

10.30385 

10.04789 

9.95211 

35 

20 

44 

26 

64851 

35149 

69647 

30353 

04796 

95204 

34 

16 

48 

27 

64877 

35123 

69679 

30321 

04802 

95198 

33 

12 

52 

28 

64902 

35098 

69710 

30290 

04808 

95192 

32 

8 

56 

29 

64927 

35073 

69742 

30258 

04S15 

95185 

31 

4 

46 

30 

9.64953 

10.35047 

9.69774 

10.30226 

10.04821 

9.95179 

30 

14: 

4 

31 

64978 

35022 

69805 

30195 

04827 

95173 

29 

50 

8 

32 

65003 

34997 

69837 

30163 

04833 

95167 

28 

52 

12 

33 

65029 

34971 

69868 

30132 

04840 

95160 

27 

48 

16 

34 

65054 

34946 

69900 

30100 

04846 

95154 

26 

44 

20 

35 

9.65079 

10.34921 

9.69932 

10.30068 

10.04852 

9.95148 

25 

40 

24 

36 

65104 

34896 

69963 

30037 

04859 

95141 

24 

36 

28 

37 

65130 

34870 

69995 

30005 

04865 

95135 

23 

32 

32 

38 

65155 

34845 

70026 

29974 

04871 

95129 

22 

28 

36 

39 

65180 

34820 

7005S 

29942 

04878 

95122 

21 

24 

40 

40 

9.65205 

10.34795 

9.70089 

10.29911 

10.04884 

9.95116 

20 

20 

44 

41 

65230 

34770 

70121 

29879 

04890 

95110 

19 

16 

48 

42 

65255 

34745 

70152 

29848 

04897 

95103 

18 

12 

52 

43 

65281 

31719 

70184 

29816 

04903 

95097 

17 

8 

56 

44 

65306 

34694 

70215 

29785 

04910 

95090 

16 

4 

47 

45 

9.65331 

10.34669 

9.70247 

10.29753 

10.04916 

9.95084 

15 

13 

4 

46 

65356 

34644 

70278 

29722 

04922 

95078 

14 

56 

8 

47 

653S1 

34619 

70309 

29691 

04929 

95071 

13 

52 

12 

48 

65406 

34594 

70311 

29659 

04935 

95065 

12 

48 

16 

49 

65431 

34569 

70372 

29628 

04941 

95059 

It 

44 

20 

50 

9.65456 

10.34544 

9.70404 

10.29596 

10.04948 

9.95052 

10 

40 

24 

51 

65481 

34519 

70435 

29565 

04954 

95046 

9 

36 

28 

52 

65506 

34494 

70466 

29534 

04961 

95039 

8 

32 

32 

53 

65531 

34469 

70498 

29502 

04967 

95033 

7 

28 

36 

54 

65556 

34441 

70529 

29471 

04973 

95027 

6 

24 

40 

55 

9.65580 

10.34420 

9.70560 

10.29440 

10.04980 

9.95020 

5 

20 

44 

56 

65605 

34395 

70592 

29408 

04986 

95014 

4 

16 

48 

57 

65630 

34370 

70623 

29377 

04993 

95007 

3 

12 

52 

58 

65655 

34345 

70654 

29346 

04999 

95001 

2 

8 

56 

59 

65680 

34320 

70685 

29315 

05005 

94995 

1 

4 

48 

60 

65705 

34295 

70717 

29283 

05012 

949:8 

0 

ir-3 

M.S. 

M 1 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Coseoant. 

Sine. 

M j 

M.S. 

?h 

116° 






OS 

cc 

o 

4 h 






















190 Logarithms Trigonometric. 



27° 



Logarithms. 


152° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

48 

0 

9.65705 

10.34295 

9.70717 

10.29283 

10.05012 

9.94988 

60 

13 

4 

i 

65729 

34271 

70748 

29252 

05018 

94982 

59 

56 

8 

2 

65754 

34246 

70779 

29221 

05025 

94975 

58 

52 

12 

3 

65779 

34221 

70810 

29190 

05031 

94969 

57 

48 

16 

4 

65804 

34196 

70841 

29159 

05038 

94962 

56 

44 

20 

5 

9.65828 

10.34172 

9.70873 

10.29127 

10.05044 

9.94956 

55 

40 

24 

6 

65853 

34147 

70904 

29096 

05051 

94949 

54 

36 

28 

7 

65878 

34122 

70935 

29065 

- 05057 

94943 

53 

32 

32 

8 

65902 

34098 

70906 

29034 

05064 

94936 

52 

28 

36 

9 

65927 

34073 

70997 

29003 

05070 

94930 

51 

24 

40 

10 

9.65952 

10.34018 

9.71028 

10.28972 

10.05077 

9.94923 

50 

20 

44 

11 

65976 

34024 

71059 

28941 

05083 

94917 

49 

16 

48 

12 

66001 

33999 

71090 

28910 

05089 

94911 

48 

12 

62 

13 

66025 

33975 

71121 

28879 

05096 

94904 

47 

8 

56 

14 

66050 

33950 

71163 

28847 

05102 

94898 

46 

4 

49 

15 

9.66075 

10.33925 

9.71184 

10.28816 

10.05109 

9.94891 

45 

11 

4 

16 

66099 

33901 

71215 

28785 

05115 

94885 

44 

56 

8 

17 

66124 

33876 

71246 

28751 

05122 

94878 

43 

52 

12 

18 

66148 

33852 

71277 

28723 

05129 

94871 

42 

48 

16 

19 

66173 

33827 

71308 

28692 

05135 

94865 

41 

44 

20 

20 

9.66197 

10.33803 

9.71339 

10.28661 

10.05142 

9.94858 

40 

40 

24 

21 

66221 

33779 

71370 

28630 

05148 

94852 

39 

36 

28 

22 

66246 

33754 

71401 

28599 

05155 

91845 

38 

32 

32 

23 

66270 

33730 

71431 

28569 

05161 

94839 

37 

28 

36 

24 

66295 

33705 

71462 

28538 

05168 

94832 

36 

24 

40 

25 

9.66319 

10.33681 

9.71493 

10.28507 

10.05174 

9.94826 

35 

20 

44 

26 

66343 

33657 

71524 

28476 

05181 

94819 

34 

16 

48 

27 

66368 

33632 

71555 

28445 

05187 

94813 

33 

12 

52 

28 

66392 

33608 

71586 

28414 

05194 

94806 

32 

8 

56 

29 

66416 

33584 

71617 

28383 

05201 

94799 

31 

4 

50 

30 

9.66441 

10.33559 

9.71648 

10.28352 

10.05207 

9.94793 

30 

10 

4 

31 

66465 

33535 

71679 

28321 

05214 

94786 

29 

56 

8 

32 

66489 

3:3511 

71709 

28291 

05220 

94780 

28 

52 

12 

33 

66513 

33487 

71740 

28260 

05227 

94773 

27 

48 

16 

34 

66537 

33463 

7177 4 

28229 

05233 

94767 

26 

44 

20 

35 

9.66562 

10.33438 

9.71802 

10.28198 

10.05240 

9.94760 

25 

40 

24 

36 

66586 

33414 

71833 

28167 

05247 

94753 

24 

36 

28 

37 

66610 

33390 

71863 

28137 

05253 

94747 

23 

32 

32 

38 

66634 

33366 

71894 

28108 

05260 

94740 

22 

28 

36 

39 

66658 

33342 

71925 

28075 

05266 

94734 

21 

24 

40 

40 

9.66682 

10.33318 

9.71955 

10.28045 

10.05273 

9.94727 

20 

20 

44 

41 

66706 

33294 

71986 

28014 

05280 

94720 

19 

16 

48 

42 

66731 

33269 

72017 

27983 

05286 

94714 

18 

12 

52 

43 

66755 

33245 

72048 

27952 

05293 

94707 

17 

8 

56 

44 

66779 

33221 

72078 

27922 

05300 

94700 

16 

4 

51 

45 

9.66803 

10.33197 

9.72109 

10.27891 

10.05306 

9.94694 

15 

9 

4 

46 

66827 

33173 

72140 

27860 

05313 

94637 

14 

56 

8 

47 

66851 

33149 

72170 

27830 

05320 

94680 

13 

52 

12 

48 

66875 

33125 

72201 

27799 

05326 

94674 

12 

48 

16 

49 

66899 

33101 

72231 

27769 

05333 

94667 

11 

44 

20 

50 

9.66922 

10.33078 

9.72262 

10.27738 

10.05310 

9.94660 

10 

40 

24 

51 

66946 

33054 

72293 

27707 

05348 

94654 

9 

36 

28 

52 

66970 

33030 

72323 

27677 

05353 

94647 

8 

32 

32 

53 

66994 

33006 

72354 

27646 

05360 

94640 

7 

28 

36 

54 

67018 

329S2 

72384 

27616 

05366 

94634 

6 

24 

40 

55 

9.67042 

10.32958 

9.72415 

10.27585 

10.05373 

9.94627 

5 

20 

44 

56 

67066 

32934 

72445 

27555 

05380 

94620 

4 

16 

48 

57 

67090 

32910 

72476 

27524 

05386 

94614 

3 

12 

52 

58 

67113 

32887 

72506 

27494 

05393 

94607 

2 

8 

56 

59 

67137 

32863 

72537 

274G3 

05400 

94600 

1 

4 

5:4 

60 

67161 

32839 

72567 

27433 

05407 

94593 

0 

8 

M.S. 

7 h 1 

M 1 

117° 

Cosine. 1 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

32° 

VI. s. 

4“ 



























Logarithms Trigonometric. 191 


l h 

o 

O0 



Logarithms. 


151° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

53 

0 

9.67161 

10.32839 

9.72567 

10.27433 

10.05407 

9.94593 

60 

8 

4 

1 

67185 

32815 

72598 

27402 

05413 

94587 

59 

56 

8 

2 

67208 

32792 

72628 

27372 

05420 

94580 

58 

52 

12 

3 

67232 

32768 

72659 

27341 

05427 

94573 

57 

48 

16 

4 

67256 

32744 

72689 

27311 

05433 

94567 

56 

44 

20 

5 

9.67280 

10.32720 

9.72720 

10.27280 

10.05440 

9.94560 

55 

40 

24 

6 

67303 

32697 

72750 

27250 

05447 

94553 

54 

36 

28 

7 

67327 

32673 

72780 

27220 

05454 

94546 

53 

32 

32 

8 

67350 

32650 

72811 

27189 

05460 

94540 

52 

28 

36 

9 

67374 

32626 

72841 

27159 

05467 

94533 

51 

24 

40 

10 

9.67398 

10.32602 

9.72872 

10.27128 

10.05474 

9.94526 

50 

20 

44 

11 

67421 

32579 

72902 

27098 

05481 

94519 

49 

16 

48 

12 

67445 

32555 

72932 

27068 

05487 

94513 

48 

12 

52 

13 

67468 

32532 

72963 

27037 

05494 

94506 

47 

8 

56 

14 

67492 

32508 

72993 

27007 

05501 

94499 

46 

4 

53 

15 

9.67515 

10.32485 

9.73023 

10.26977 

10.05508 

9.94492 

45 

7 

4 

16 

67539 

32461 

73054 

26946 

05515 

94485 

44 

56 

8 

17 

67562 

32438 

73084 

26916 

05521 

94479 

43 

52 

12 

18 

67586 

32414 

73114 

26886 

05528 

94472 

42 

48 

16 

19 

67609 

32391 

73144 

26856 

05535 

94465 

41 

44 

20 

20 

9.67633 

10.32367 

9.73175 

10.26825 

10.05542 

9.94458 

40 

40 

24 

21 

67656 

32344 

73205 

26795 

05549 

94451 

39 

36 

28 

22 

67680 

32320 

73235 

26765 

05555 

94445 

38 

32 

32 

23 

67703 

32297 

73265 

26735 

05562 

94438 

37 

28 

36 

24 

67726 

32274 

73295 

26705 

05569 

94431 

36 

24 

40 

25 

9.67750 

10.32250 

9.73326 

10.26674 

10.05576 

9.94424 

35 

20 

44 

26 

67773 

32227 

73356 

26644 

05583 

94417 

34 

16 

48 

27 

67796 

32204 

73386 

26614 

05590 

94410 

33 

12 

52 

28 

67820 

32180 

73416 

26584 

05596 

94404 

32 

8 

56 

29 

67843 

32157 

73446 

26554 

05603 

94397 

31 

4 

54 

30 

9.67866 

10.32134 

9.73476 

10.26524 

10.05610 

9.94390 

30 

G 

4 

31 

67890 

32110 

73507 

26493 

05617 

94383 

29 

56 

8 

32 

67913 

32087 

73537 

26463 

05624 

94376 

28 

52 

12 

33 

66936 

32064 

73567 

26433 

05631 

94369 

27 

48 

16 

34 

67959 

32041 

73597 

26403 

05638 

94362 

26 

44 

20 

35 

9.07982 

10.32018 

9.73627 

10.26373 

10.05645 

9.94355 

25 

40 

24 

36 

68006 

31994 

73657 

26343 

05651 

94349 

24 

36 

28 

37 

68029 

31971 

73687 

26313 

05658 

94342 

23 

32 

32 

38 

68052 

31948 

73717 

26283 

05665 

94335 

22 

28 

36 

39 

68075 

31925 

73747 

26253 

05672 

9432S 

21 

24 

40 

40 

9.68098 

10.31902 

9.73777 

10.26223 

10.05679 

9.94321 

20 

20 

44 

41 

68121 

31879 

73S07 

26193 

05686 

94314 

19 

16 

48 

42 

68144 

31856 

73837 

26163 

05693 

94307 

18 

12 

52 

43 

68167 

31833 

73S07 

26133 

05700 

94300 

17 

8 

56 

44 

68190 

31810 

73897 

26103 

05707 

94293 

16 

' 4 

55 

45 

9.68213 

10.31787 

9.73927 

10.26073 

10.05714 

9.94286 

15 

5 

4 

46 

68237 

31763 

73957 

26043 

05721 

94279 

14 

56 

8 

47 

68260 

31740 

73987 

26013 

05727 

94273 

13 

52 

12 

48 

68283 

31717 

74017 

25983 

05734 

94266 

12 

48 

16 

49 

68305 

31695 

74047 

25953 

05741 

94259 

11 

44 

20 

50 

9.68328 

10.31672 

9.74077 

10.25923 

10.05748 

9.94252 

10 

40 

24 

51 

68351 

31649 

74107 

25893 

05755 

94245 

9 

36 

28 

52 

68374 

31626 

74137 

25863 

05762 

94238 

8 

32 

32 

53 

68397 

31603 

74166 

25834 

05769 

94231 

7 

28 

36 

54 

68420 

31580 

74196 

25804 

05776 

94224 

6 

24 

40 

55 

9.68443 

10.31557 

9.74226 

10.25774 

10.05783 

9.94217 

5 

20 

44 

56 

68466 

31534 

74256 

25744 

05790 

■94210 

4 

16 

48 

57 

68489 

31511 

74286 

25714 

05797 

94203 

3 

12 

52 

58 

6S512 

31488 

74316 

25684 

05804 

94196 

2 

8 

56 

59 

68534 

31466 

74345 

25655 

05811 

94189 

1 

4 

56 

60 

68557 

31443 

74375 

25625 

05818 

94182 

0 

4 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M. S. 

7 h | 

118 

3 






61° 

4 U 























192 Logakithms Trigonometric. 


1“ 

29° 



Logarithms. 


150° 

10“ 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

56 

0 

9.68557 

10.31443 

9.74375 

10.25625 

10.05818 

9.94182 

60 

4 

4 

1 

68580 

31420 

74405 

25595 

05825 

94175 

59 

56 

8 

2 

68603 

31397 

74435 

25565 

05832 

94168 

58 

52 

12 

3 

68625 

31375 

74465 

25535 

05839 

94161 

57 

48 

16 

4 

68648 

31352 

74494 

25506 

05846 

94154 

56 

44 

20 

5 

9.68671 

10.31329 

9.74524 

10.25476 

10.05853 

9.94147 

55 

40 

24 

6 

68694 

31306 

74554 

25446 

05S60 

94140 

54 

36 

28 

7 

68716 

31284 

74583 

25417 

05867 

94133 

53 

32 

32 

8 

68739 

31261 

74613 

25387 

05874 

94126 

52 

28 

36 

9 

68762 

31238 

74643 

26357 

05881 

94119 

51 

24 

40 

10 

9.68784 

10.31216 

9.74673 

10.25327 

10.05888 

9.94112 

50 

20 

44 

11 

68807 

31193 

74702 

25298 

05895 

94105 

49 

16 

48 

12 

68829 

31171 

74732 

25268 

05902 

94098 

48 

12 

52 

13 

6S852 

31148 

74762 

25238 

05910 

91090 

47 

8 

56 

14 

68875 

31125 

74791 

25209 

05917 

94083 

46 

4 

57 

15 

9.68897 

10.31103 

9.74821 

10.25179 

10.05924 

9.94076 

45 

3 

4 

16 

68920 

31080 

74851 

25149 

05931 

94069 

44 

56 

8 

17 

68942 

31058 

74880 

25120 

05938 

94062 

43 

52 

12 

18 

68965 

31035 

74910 

25090 

05945 

94055 

42 

48 

16 

19 

68987 

31013 

74939 

25061 

05952 

9404S 

41 

44 

20 

20 

9.69010 

10.30990 

9.74969 

10.25031 

10.05959 

9.94041 

40 

40 

24 

21 

69032 

30968 

74998 

25002 

05966 

94031 

39 

36 

28 

22 

69055 

30945 

75028 

24972 

05973 

94027 * 

38 

32 

32 

23 

69077 

30923 

75058 

24942 

05980 

94020 

37 

28 

36 

24 

69100 

30900 

75087 

24913 

05988 

94012 

36 

24 

40 

25 

9.69122 

10.30878 

9.75117 

10.24883 

10.05995 

9.94005 

35 

20 

44 

26 

69144 

30856 

75146 

24854 

06002 

93998 

34 

16 

48 

27 

69167 

30S33 

75176 

24824 

06009 

93991 

33 

12 

52 

28 

691S9 

30811 

75205 

24795 

06016 

93984 

32 

8 

56 

29 

69212 

30788 

75235 

24765 

06023 

93977 

31 

4 

58 

30 

9.69234 

10.30766 

9.75264 

10.24736 

10.06030 

9.93970 

30 

2 

4 

31 

69256 

30744 

75294 

24706 

06037 

93963 

29 

56 

8 

32 

69279 

30721 

75323 

24677 

06045 

93955 

28 

52 

12 

33 

69301 

30699 

75353 

24647 

06052 

93948 

27 

48 

16 

34 

69323 

30677 

75382 

24618 

- 06059 

93941 

26 

44 

20 

35 

9.69345 

10.30655 

9.75411 

10.24589 

10.06066 

9.93934 

25 

40 

24 

36 

69368 

30632 

75441 

24559 

06073 

93927 

24 

36 

28 

37 

69390 

30610 

75470 

24530 

06080 

93920 

23 

32 

32 

38 

69412 

30588 

75500 

24500 

06088 

93912 

22 

28 

36 

39 

69434 

30566 

75529 

24471 

06095 

93905 

21 

24 

40 

40 

9.69456 

10.30544 

9.75558 

10.24442 

10.06102 

9.93S98 

20 

20 

44 

41 

69479 

30521 

75588 

24412 

06109 

93891 

19 

16 

48 

42 

69501 

30499 

75617 

24383 

06116 

93884 

18 

12 

52. 

43 

69523 

30477 

75647 

24353 

06124 

93876 

17 

8 

56 

44 

69545 

30455 

75676 

24324 

06131 

93869 

16 

4 

59 

45 

9.69567 

10.30433 

9.75705 

10.24295 

10.06138 

9.93862 

15 

1 

4 

46 

69589 

30411 

75735 

24265 

06145 

93855 

14 

56 

8 

47 

69611 

30389 

75764 

24236 

06153 

93847 

13 

52 

12 

48 

69633 

30367 

75793 

24207 

•06160 

93840 

12 

48 

16 

49 

69655 

30345 

75822 

24178 

06167 

'93833 

11 

44 

20 

50 

9.69677 

10.30323 

9.75852 

10.24148 

10.06174 

9.93826 

10 

40 

24 

51 

69699 

30301 

75S81 

24119 

06181 

93619 

9 

36 

28 

52 

69721 

30279 

75910 

24090 

06189 

93811 

8 

32 

32 

53 

69743 

30257 

75939 

21061 

06196 

93804 

7 

28 

36 

54 

69765 

30235 

75969 

24031 

06203 

93797 

6 

24 

40 

55 

9.69787 

10.30213 

9.75998 

10.24002 

10.06211 

9.93789 

5 

20 

44 

56 

69809 

30191 

76027 

23973 

062 IS 

93782 

4 

16 

48 

57 

69831 

30169 

76056 

23944 

06225 

93775 

3 

12 

52 

58 

69853 

30147 

76086 

23914 

06232 

93763 

2 

8 

56 

59 

69875 

30125 

76115 

238S5 

06240 

93760 

1 

4 

66 

60 

69897 

30103 

76144 

23856 

06247 

93753 

0 

O 

M. S. 

7 “ 

11 

119 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

J1 

60° 

M. S. 

4“ 






















Logarithms Trigonometric. 


193 


Logarithms. 


Tangent. 


Cotangent. 


2 h 

0 

o 

CO 

M.S. 

M 

0 

0 

4 

1 

8 

2 

12 

3 

16 

4 

20 

5 

24 

6 

28 

7 

32 

8 

36 

9 

40 

10 

44 

11 

48 

12 

'52 

13 

56 

14 

1 

15 

4 

16 

8 

17 

12 

18 

16 

19 

20 

20 

24 

21 

28 

22 

32 

23 

36 

24 

40 

25 

44 

26 

48 

27 

52 

28 

56 

29 

3 

30 

4 

31 

8 

32 

12 

33 

16 

34 

20 

35 

24 

36 

28 

37 

32 

38 

36 

39 

40 

40 

44 

41 

48 

42 

52 

43 

56 

44 

3 

45 

4 

46 

8 

47 

12 

48 

16 

49 

20 

50 

24 

51 

28 

52 

32 

53 

36 

51 

40 

55 

44 

56 

4S 

57 

52 

5S 

56 

59 

4 

60 

M.S. 

M 

8 h 

120 


Sine. 

9.G9897 

69919 

69941 

69953 

69984 

9.70006 

70028 

70050 

70072 

70093 

9.70115 

70137 

70159 

70180 

70202 

9.70224 

70245 

70267 

70288 

70310 

9.70332 

70353 

70375 

70396 

70418 

9.70439 

70461 

70482 

70504 

70525 

9.70547 

70568 

70590 

70611 

70633 

9.70654 

70675 

70697 

70718 

70739 

9.70761 

70782 

70803 

70824 

70S46 

9.70867 

70888 

70909 

70931 

70952 

9.70973 

70994 

71015 

71036 

71058 

9.71079 

71100 

71121 

71142 

71163 

71184 

Cosine. 


Cosecant. 

10.30103 

30081 

30059 

30037 

30016 

10.29994 

29972 

29950 

29928 

29907 

10.29885 

29863 

29841 

29820 

29798 

10.29776 

29755 

29733 

29712 

29G90 

10.29668 

29647 

29625 

29604 

29582 

10.29561 

29539 

29518 

29496 

29475 

10.29453 

29432 

29410 

29389 

29367 

10.29346 

29325 

29303 

29282 

29261 

10.29239 

29218 

29197 

29176 

29151 

10.29133 

29112 

29091 

29069 

29048 

10.29027 

29006 

28985 

28964 

28942 

10.28921 

28900 

28879 

28858 

28837 

28816 

Secant. 


9.76144 
76173 
76202 
76231 
76261 
9.76290 
76319 
76348 
76377 
76406 
9.76435 
76464 
76493 
76522 
76551 
9.76580 
76609 
76639 
76668 
76697 
9.76725 
76754 
76783 
76812 
76841 
9.76870 
76899 
76928 
76957 
76986 
9.77015 
77044 
77073 
77101 
77130 
9.77159 
77188 
77217 
77246 
77274 
9.77303 
77332 
77361 
77390 
77418 
9.77447 
77476 
77505 
77533 
77562 
9.77591 
77619 
77648 
77677 
77706 
9.77734 
77763 
' 77791 
77820 
77849 
77877 

Cotangent 


10.23856 

23827 

23798 

23769 

23739 

10.23710 

23681 

23652 

23623 

23594 

10.23565 

23536 

23507 

23478 

23449 

10.23420 

23391 

23361 

23332 

23303 

10.23275 

23246 

23217 

23188 

23159 

10.23130 

23101 

23072 

23043 

23014 

10.22985 

22956 

22927 

22899 

22870 

10.22841 

22812 

22783 

22754 

22726 

10.22697 

22668 

22639 

22610 

22582 

10.22553 

22524 

22495 

22467 

22438 

10.22409 

22381 

22352 

22323 

22294 

10.22266 

22237 

22209 

22180 

22151 

22123 

Tangent. 


Secant. 

10.06247 

06254 

06262 

06269 

06276 

10.06283 

06291 

06298 

06305 

06313 

10.06320 

06327 

06335 

06342 

06350 

10.06357 

06364 

06372 

06379 

063S6 

10.06394 

06401 

06409 

06416 

06423 

10.06431 

06438 

06446 

06453 

06461 

10.06468 

06475 

06483 

06490 

06498 

10.06505 

06513 

06520 

06528 

06535 

10.06543 

06550 

06558 

06565 

06573 

10.06580 

06588 

06595 

06603 

06610 

10.06618 

06525 

06633 

06640 

06648 

10.06656 

06663 

06671 

06678 

06686 

06693 

Cosecant. 


Cosine. 

9.93753 

93746 

93738 

93731 

93724 

9.93717 

93709 

93702 

93695 

93687 

9.93680 

93673 

93665 

93658 

93650 

9.93643 

93636 

93628 

93621 

93614 

9.93606 

93599 

93591 

93584 

93577 

9.93569 

93562 

93554 

93547 

93539 

9.93532 

93525 

93517 

93510 

93502 

9.93495 

93487 

93480 

93472 

93465 

9.93457 

93450 

93442 

93435 

93427 

9.93420 

93412 

93405 

93397 

93390 

9.93382 

93375 

93367 

93360 

93352 

9.93344 

933.37 

93329 

93322 

93314 

93307 

Sine. 


49° 

9 h 

M 

M.S. 

60 

GO 

59 

56 

58 

52 

57 

48 

56 

44 

55 

40 

54 

36 

53 

32 

52 

28 

51 

24 

50 

20 

49 

16 

48 

12 

47 

8 

46 

4 

45 

59 

44 

56 

43 

52 

42 

48 

41 

44 

40 

40 

39 

36 

38 

32 

37 

28 

36 

24 

35 

20 

34 

16 

33 

12 

32 

8 

31 

4 

30 

58 

29 

56 

28 

52 

27 

48 

26 

44 

25 

40 

24 

36 

23 

32 

22 

28 

21 

24 

20 

20 

19 

16 

18 

12 

17 

8 

16 

4 

15 

57 

14 

56 

13 

52 

12 

48 

11 

44 

10 

40 

9 

36 

8 

32 

7 

28 

6 

24 

5 

20 

4 

16 

3 

12 

2 

8 

1 

4 

0 

56 

M 

M. S. 

59° 

3 h 


13 



















If 4 Logarithms Trigonometric. 


2 h 

31° 



Logarithms. 


148° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

1 M 

M.S. 

4- 

0 

9.71184 

10.28816 

9.77877 

10.22123 

10.06693 

9.93307 

60 

56 

4 

1 

71205 

28795 

77906 

22094 

06701 

93299 

59 

56 

8 

2 

71226 

28774 

77935 

22065 

06709 

93291 

58 

52 

12 

3 

71247 

28753 

77963 

22037 

06716 

93284 

57 

48 

16 

4 

71268 

28732 

77992 

22008 

06724 

93276 

56 

44 

20 

5 

9.71289 

10.28711 

9.78020 

10.21980 

10.06731 

9.93269 

55 

40 

24 

6 

71310 

28690 

78049 

21951 

06739 

93261 

54 

36 

28 

7 

71331 

28669 

78077 

21923 

06747 

93253 

53 

32 

32 

8 

71352 

2864S 

78106 

21894 

06754 

93246 

52 

28 

36 

9 

71373 

28627 

78135 

21865 

06762 

93238 

51 

24 

40 

10 

9.71393 

10.28607 

9.78163 

10.21837 

10.06770 

9.93230 

50 

20 

44 

11 

71414 

28586 

78192 

21808 

06777 

93223 

49 

16 

48 

12 

71435 

28565 

78220 

21780 

06785 

93215 

48 

12 

52 

13 

71456 

28544 

78249 

21751 

06793 

93207 

47 

8 

56 

14 

71477 

28523 

78277 

21723 

06800 

93200 

46 

4 

5 

15 

9.71498 

10.28502 

9.78306 

10.21694 

10.06808 

9.93192 

45 

55 

4 

16 

71519 

28481 

78334 

21666 

06816 

93184 

44 

56 

8 

17 

71539 

28461 

78363 

21637 

06823 

93177 

43 

52 

12 

18 

71560 

28440 

78391 

21609 

06831 

93169 

42 

48 

16 

19 

71581 

28419 

78419 

21581 

06839 

93161 

41 

44 

20 

20 

9.71602 

10.28398 

9.78448 

10.21552 

10.06846 

9.93154 

40 

40 

24 

21 

71622 

23378 

78476 

21524 

06854 

93146 

39 

36 

28 

22 

71643 

28357 

78505 

21495 

06862 

93138 

38 

32 

32 

23 

71664 

28336 

78533 

21467 

06869 

93131 

37 

28 

36 

24 

71685 

28315 

78562 

21438 

06877 

93123 

36 

24 

40 

25 

9.71705 

10.28295 

9.78590 

10.21410 

10.06885 

9.93115 

35 

20 

44 

26 

71726 

28274 

78618 

21382 

06892 

93108 

34 

16 

48 

27 

71747 

28253 

78647 

21353 

06900 

93100 

33 

12 

52 

28 

71767 

28233 

78675 

21325 

06908 

93092 

32 

8 

56 

29 

71788 

28212 

78704 

21296 

06916 

93084 

31 

4 

6 

30 

9.71809 

10.28191 

9.78732 

10.21268 

10.06923 

9.93077 

30 

54 

4 

31 

71829 

28171 

78760 

21240 

06931 

93069 

29 

56 

8 

32 

71850 

28150 

78789 

21211 

06939 

93061 

28 

52 

12 

33 

71870 

28130 

78817 

21183 

06947 

93053 

27 

48 

16 

34 

71891 

28109 

78845 

21155 

06954 

93046 

26 

44 

20 

35 

9.71911 

10.28089 

9.78874 

10.21126 

10.06962 

9.9303S 

25 

40 

24 

36 

71932 

28068 

78902 

21098 

06970 

93030 

24 

36 

28 

37 

71952 

28048 

78930 

21070 

06978 

93022 

23 

32 

32 

38 

71973 

28027 

78959 

21041 

06986 

93014 

22 

28 

36 

39 

71994 

28006 

78987 

21013 

06993 

93007 

21 

24 

40 

40 

9.72014 

10.27986 

9.79015 

10.20985 

10.07001 

9.92999 

20 

20 

44 

41 

72034 

27966 

79043 

20957 

07009 

92991 

19 

16 

48 

42 

72055 

27945 

79072 

20928 

07017 

92983 

18 

12 

52 

43 

72075 

27925 

79100 

20900 

07024 

92976 

17 

8 

56 

44 

72096 

27904 

79128 

20872 

07032 

92968 

16 

4 

7 

45 

9.72116 

10.27884 

9.79156 

10.20844 

10.07040 

9.92960 

15 

53 

4 

46 

72137 

27863 

79185 

20815 

07048 

92952 

14 

56 

8 

47 

72157 

27843 

79213 

20787 

07056 

13944 

13 

52 

12 

48 

72177 

27823 

79241 

20759 

07064 

93936 

12 

18 

16 

49 

72198 

27802 

79269 

20731 

07071 

92929 

11 

44 

20 

50 

9.72218 

10.27782 

9.79297 

10.20703 

10.07079 

9.92921 

10 

40 

24 

51 

72238 

27762 

79326 

20674 

07087 

92913 

9 

36 

28 

52 

72259 

27741 

79354 

20646 

07095 

92905 

8 

32 

32 

53 

72279 

27721 

79382 

20618 

07103 

92897 

7 

28 

36 

54 

72299 

27701 

79410 

20590 

07111 

92889 

6 

24 

40 

55 

9.72320 

10.27680 

9.79438 

10.20562 

10.07119 

9.92881 

5 

20 

44 

56 

72340 

27660 

79466 

20534 

07126 

92874 

4 

16 

48 

57 

72360 

27640 

79495 

20505 

07134 

92866 

3 

12 

52 

58 

72381 

27619 

79523 

20477 

07142 

92858 

2 

8 

56 

59 

72401 

27599 

79551 

20449 

07150 

92850 

1 

4 

8 

60 

72421 

27579 

79579 

20421 

07158 

92842 

0 

5a 

M.S. 

8 b 

M 

121 ( 

Cosiue. 

5 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

58° 

M.S. 

3 b 






















Logarithms Trigonometric. 195 


2 h 

32° 


Logarithms. 


147 c 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

8 

0 

9.72121 

10.27579 

9.79579 

10.20421 

10.07158 

9.92842 

60 

53 

4 

1 

72411 

27559 

79607 

20393 

07166 

92834 

59 

56 

8 

2 

72161 

27539 

79635 

20365 

07174 

92826 

58 

52 

12 

3 

72482 

27518 

79663 

20337 

07182 

92818 

57 

48 

10 

4 

72502 

27498 

79691 

20309 

07190 

92810 

56 

44 

20 

5 

9.72522 

10.27478 

9.79719 

10.20281 

10.07197 

9.92803 

55 

40 

24 

6 

72512 

27458 

79747 

20253 

07205 

92795 

54 

36 

28 

7 

72562 

27438 

79776 

20224 

07213 

92787 

63 

32 

32 

8 

72582 

27418 

79804 

20196 

07221 

92779 

62 

28 

36 

9 

72602 

27398 

79832 

2016S 

07229 

92771 

51 

24 

40 

10 

9.72622 

10.27378 

9.79860 

10.20140 

10.07237 

9.92763 

50 

20 

44 

11 

72643 

27357 

79888 

20112 

07245 

92755 

49 

16 

48 

12 

72663 

27337 

79916 

20084 

07253 

92747 

48 

12 

52 

13 

72683 

27317 

79944 

20056 

07261 

92739 

47 

8 

50 

14 

72703 

27297 

79972 

20028 

07269 

92731 

46 

4 

‘J 

15 

9.72723 

10.27277 

9.80000 

10.20000 

10.07277 

9.92723 

,45 

51 

4 

16 

72743 

27267 

80028 

19972 

07285 

92715 

44 

56 

8 

17 

72763 

27237 

80056 

19944 

07293 

92707 

43 

52 

12 

18 

72783 

27217 

800S4 

19916 

07301 

92699 

42 

4S 

16 

19 

72803 

27197 

80112 

19888 

07309 

92691 

41 

44 

20 

20 

9.72823 

10.27177 

9.80140 

10.19860 

10.07317 

9.92683 

40 

40 

24 

21 

72843 

27157 

80168 

19832 

07325 

92675 

39 

36 

28 

22 

72S63 

27137 

80195 

19805 

07333 

92667 

38 

32 

32 

23 

72883 

27117 

80223 

19777 

07341 

92659 

37 

28 

36 

24 

72902 

27098 

80251 

19749 

07349 

92651 

36 

24 

40 

25 

9.72922 

10.27078 

9.80279 

10.19721 

10.07357 

9.92643 

35 

20 

44 

26 

72942 

27058 

80307 

19693 

07365 

92635 

34 

16 

48 

27 

72902 

27038 

80335 

19665 

07373 

92627 

33 

12 

52 

28 

72982 

27018 

80363 

19637 

07381 

92619 

32 

8 

56 

29 

73002 

26998 

80391 

19609 

07389 

92611 

31 

4 

10 

30 

9.73022 

10.26978 

9.80419 

10.19581 

10.07397 

9.92603 

30 

50 

4 

31 

73041 

26959 

80447 

19553 

07405 

92595 

29 

56 

8 

32 

■73061 

20939 

80474 

19526 

07413 

92587 

28 

52 

12 

33 

73081 

26919 

80502 

19498 

07121 

92579 

27 

48 

16 

34 

73101 

26899 

80530 

19470 

07429 

92571 

26 

44 

20 

35 

9.73121 

10.26879 

9.S0558 

10.19442 

10.07437 

9.92563 

25 

40 

24 

36 

73140 

26860 

805S6 

19414 

07445 

92555 

24 

36 

28 

37 

73160 

26840 

80614 

19386 

07454 

92546 

23 

32 

32 

38 

73180 

26820 

80642 

19358 

07462 

92538 

22 

28 

36 

39 

732i tO 

26S00 

80669 

19331 

07470 

92530 

21 

24 

40 

40 

9.73219 

10.26781 

9.80697 

10.19303 

10.07478 

9.92522 

20 

20 

44 

41 

73239 

26761 

80725 

19275 

074S6 

92514 

19 

16 

48 

42 

73259 

26741 

80753 

19247 

07494 

92506 

18 

12 

52 

43 

73278 

26722 

80781 

19219 

07502 

92498 

17 

8 

56 

44 

73298 

26702 

80808 

19192 

07510 

92490 

16 

4 

11 

45 

9.73318 

10.26682 

9.80836 

10.19164 

10.07518 

9.92482 

15 

49 

4 

46 

73337 

26663 

80864 

19136 

07527 

92473 

14 

56 

8 

47 

73357 

26643 

80892 

19108 

07535 

92465 

13 

52 

12 

48 

73377 

26623 

80919 

19081 

C7543 

92457 

12 

48 

16 

49 

73396 

26604 

80947 

19053 

07551 

92449 

11 

44 

20 

50 

9.73416 

10.26584 

9.80975 

10.19025 

10.07559 

9.92441 

10 

40 

24 

51 

73435 

26565 

81003 

18997 

07567 

92433 

9 

36 

28 

52 

73455 

26545 

81030 

18970 

07575 

92425 

8 

32 

32 

53 

73471 

26526 

81058 

18942 

075S4 

92416 

7 

28 

36 

54 

73494 

26606 

81086 

18914 

07592 

92408 

6 

24 

40 

55 

9.73513 

1026487 

9.81113 

10.18887 

10.07600 

9.924O0 

5 

20 

44 

56 

7 3533 

26467 

81141 

18859 

07608 

92392 

4 

16 

48 

57 

73552 

26448 

81169 

18831 

07616 

92384 

3 

12 

52 

58 

73572 

26428 

81196 

18804 

07624 

92376 

2 

8 

56 

59 

73591 

26409 

81224 

18776 

07633 

92367 

1 

4 

13 

60 

73611 

26389 

81252 

18748 

07641 

92359 

0 

48 

M.S. 

8 h 

M 

122‘ 

Cosine. 

5 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

57° 

M. S. 

3 h 





















196 Logarithms Trigonometric. 


2 h 

33° 



Logarithms. 


146 c 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

1:4 

0 

9.73611 

10.26389 

9.81252 

10.18748 

10.07641 

9.92359 

60 

48 

4 

1 

73630 

26370 

81279 

18721 

07649 

92351 

59 

56 

8 

2 

73650 

26350 

81307 

18693 

07657 

92343 

58 

52 

12 

3 

73669 

26331 

81335 

18665 

07665 

92335 

57 

48 

16 

4 

73689 

26311 

81362 

18638 

07674 

92326 

56 

44 

20 

5 

9.73708 

10.26292 

9.81390 

10.18610 

10.07682 

9.92318 

55 

40 

24 

6 

73727 

’ 26273 

81418 

185S2 

07690 

92310 

54 

36 

28 

7 

73747 

26253 

81445 

18555 

07698 

92302 

53 

32 

82 

8 

73766 

26234 

81473 

18527 

07707 

92293 

52 

28 

36 

9 

73785 

26215 

81500 

18500 

07715 

92285 

51 

24 

40 

10 

9.73805 

10.26195 

9.81528 

10.18472 

10.07723 

9.92277 

50 

20 

44 

11 

73824 

26176 

81556 

18444 

07731 

92269 

49 

16 

48 

12 

73843 

26157 

81583 

18417 

07740 

92260 

48 

12 

52 

13 

73863 

26137 

81611 

18389 

07748 

92252 

47 

8 

56 

14 

73882 

26118 

81638 

18362 

07756 

92244 

46 

4 

13 

15 

9.73901 

10.26099 

9.81666 

10.18334 

10.07765 

9.92235 

45 

47 

4 

16 

73921 

26079 

81693 

18307 

07773 

92227 

44 

56 

8 

17 

73940 

26060 

81721 

18279 

07781 

92219 

43 

52 

12 

18 

73959 

26041 

81748 

18252 

07789 

92211 

42 

48 

16 

19 

73978 

26022 

81776 

18224 

07798 

92202 

41 

44 

20 

20 

9.73997 

10.26003 

9.81803 

10.18197 

10.07806 

9.92194 

40 

40 

24 

21 

74017 

25983 

81831 

18169 

07814 

92186 

39 

36 

28 

22 

74036 

25964 

81858 

18142 

07823 

92177 

38 

32 

32 

23 

74055 

25945 

81886 

18114 

07831 

92169 

37 

28 

36 

24 

74074 

25926 

81913 

18087 

07839 

92161 

36 

24 

40 

25 

9.74093 

10.25907 

9.81941 

10.18059 

10.07848 

9.92152 

35 

20 

44 

26 

74113 

258S7 

81968 

18032 

07856 

92144 

34 

16 

48 

27 

74132 

25868 

81996 

18004 

07864 

92136 

33 

12 

52 

28 

74151 

25849 

82023 

17977 

07873 

92127 

32 

8 

56 

29 

74170 

25830 

82051 

17949 

07881 

92119 

31 

4 

14 

30 

9.74189 

10.25S11 

9.82078 

10.17922 

10.07889 

9.92111 

30 

46 

4 

31 

74208 

25792 

82106 

17894 

07898 

92102 

29 

56 

8 

32 

74227 

25773 

82133 

17867 

07906 

92094 

28 

52 

12 

33 

74246 

25754 

82161 

17839 

07914 

92086 

27 

48 

16 

34 

74265 

25735 

821S8 

17812 

07923 

92077 

26 

44 

20 

35 

9.74284 

10.25716 

9.82215 

10.17785 

10.07931 

9.92069 

25 

40 

24 

36 

74303 

25697 

82243 

17757 

07940 

92060 

24 

36 

28 

37 

74322 

’ 25678 

82270 

17730 

07948 

92052 

23 

32 

32 

38 

74341 

25659 

82298 

17702 

07956 

92044 

22 

28 

36 

39 

74360 

25640 

82325 

17675 

07965 

92035 

21 

24 

40 

40 

9.74379 

10.25621 

9.82352 

10.17648 

10 07973 

9.92027 

20 

20 

44 

41 

74398 

25602 

82380 

17620 

07982 

92018 

19 

16 

48 

42 

74417 

25583 

82407 

17593 

07990 

92010 

18 

12 

52 

43 

74436 

25564 

82435 

17565 

07998 

92002 

17 

8 

56 

44 

74455 

25545 

82462 

17538 

08007 

91993 

16 

4 

15 

45 

9.74474 

10.25526 

9.82489 

10.17511 

10.08015 

9 91985 

15 

45 

4 

46 

74493 

25507 

82517 

17483 

08024 

91976 

14 

56 

8 

47 

74512 

25488 

82544 

17456 

08032 

91968 

13 

52 

12 

48 

74531 

25469 

82571 

17429 

08041 

91959 

12 

48 

16 

49 

74549 

25451 

82599 

17401 

08049 

91951' 

11 

44 

20 

50 

9.74568 

10.25432 

9.82626 

10.17374 

10.08058 

9.91942 

10 

40 

24 

51 

74587 

25413 

82653 

17347 

08U66 

91934 

9 

36 

28 

52 

74606 

25394 

82681 

77319 

08075 

91925 

8 

32 

32 

53 

74625 

25375 

82708 

17292 

08083 

91917 

l 

28 

36 

54 

74644 

25356 

82735 

17265 

08092 

91908 

6 

24 

40 

55 

9.74662 

10.25338 

9.82762 

10.17238 

10.08100 

9.91900 

5 

20 

44 

56 

74681 

25319 

82760 

17210 

08109 

91891 

4 

16 

48 

57 

74700 

25300 

82817 

17183 

08117 

91883 

3 

12 

52 

58 

74719 

25281 

82844 

17156 

08126 

91874 

2 

8 

56 

59 

74737 

25263 

82871 

17129 

08134 

91866 

1 

4 

16 

60 

74766 

25244 

82S99 

17101 

08143 

91857 

0 

44 

M.S. 

8 h 

M 

123 

Cosine. 

3 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

56° 

M. S. 

3 h 



















Logarithms Trigonometric. 197 


2 h 

34° 



Logarithms. 


145° 

9“ 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

1G 

0 

9.74756 

10.25244 

9.82899 

10.17101 

10.08143 

9.91857 

60 

44 

4 

1 

74775 

25225 

82926 

17074 

08151 

91849 

59 

56 

8 

2 

74794 

25206 

82953 

17047 

08160 

91840 

58 

52 

12 

3 

71812 

25188 

82980 

17020 

08168 

91832 

57 

48 

16 

4 

74831 

25169 

83008 

16992 

OS177 

91823 

56 

44 

20 

5 

9.74850 

10.25150 

9.83035 

10.16965 

10.08185 

9.91815 

55 

40 

24 

6 

74868 

25132 

83062 

16938 

08194 

91806 

54 

36 

28 

7 

74887 

25113 

83089 

16911 

08202 

91798 

53 

32 

32 

8 

74906 

25094 

83117 

16883 

08211 

91789 

52 

28 

36 

9 

74924 

25076 

83144 

16856 

08219 

91781 

51 

24 ' 

40 

10 

9.74943 

10.25057 

9.83171 

10.16829 

10.08228 

9.91772 

50 

20 

44 

11 

74961 

25039 

83198 

16802 

08237 

91763 

49 

16 

48 

12 

74980 

25020 

83225 

16775 

08245 

91755 

48 

12 

52 

13 

74909 

25001 

83252 

16748 

08254 

91746 

47 

8 

50 

14 

75017 

24983 

83280 

16720 

08262 

91738 

46 

4 

17 

15 

9.75030 

10.24964 

9.83307 

10.16693 

10.08271 

9.91729 

45 

43 

4 

16 

75054 

■ 24946 

83334 

16666 

08280 

91720 

44 

56 

8 

17 

75073 

24927 

83361 

16639 

08288 

91712 

43 

52 

12 

18 

7509 L 

24909 

83388 

16612 

08297' 

91703 

42 

48 

16 

19 

75110 

24896 

83415 

16585 

08305 

91695 

41 

44 

20 

20 

9.75128 

10.24872 

9.83442 

10.16558 

10.08314 

9.91686 

40 

40 

24 

21 

75147 

24853 

83470 

16530 

08323 

91677 

39 

36 

28 

22 

75165 

24835 

83497 

16503 

08331 

91669 

38 

32 

82 

23 

75184 

24816 

83524 

16476 

08340 

91660 

37 

28 

36 

24 

75202 

24798 

83551 

16449 

08349 

91651 

36 

24 

40 

25 

9.75221 

10.24779 

9.83578 

10.16422 

10.08357 

9.91643 

35 

20 

44 

26 

75239 

24761 

83605 

16395 

08366 

91634 

34 

16 

48 

27 

75258 

24742 

83632 

16368 

08375 

91625 

33 

12 

52 

28 

75276 

24724 

83659 

16341 

08383 

91617 

32 

8 

56 

29 

75291 

24706 

83686 

16314 

08392 

91608 

31 

4 

18 

30 

9.75313 

10.24687 

9.83713 

10.16287 

10.08401 

9.91599 

30 

43 

4 

31 

75331 

24669 

83740 

16260 

08409 

91591 

29 

56 

8 

32 

75350 

24650 

83768 

16232 

08418 

91582 

28 

52 

12 

33 

75368 

24632 

83795 

16205 

08427 

91573 

27 

48 

16 

34 

75386 

24614 

83822 

16178 

08435 

91565 

26 

44 

20 

35 

9.75405 

10.24595 

9.83849 

10.16151 

10.08444 

9.91556 

25 

40 

24 

36 

75423 

24577 

83876 

16124 

08153 

91547 

24 

36 

28 

37 

75441 

24559 

83903 

16097 

08462 

91538 

23 

32 

32 

38 

75459 

24541 

83930 

16070 

08470 

91530 

22 

28 

36 

39 

75478 

24522 

83957 

16043 

08479 

91521 

21 

24 

40 

40 

9.75496 

10.24504 

9.83984 

10.16016 

10.08488 

9.91512 

20 

20 

44 

41 

75514 

24486 

84011 

159S9 

08496 

91504 

19 

16 

48 

42 

75533 

24467 

84038 

15962 

08505 

91495 

18 

12 

52 

43 

75551 

24449 

84065 

15935 

08514 

91486 

17 

8 

56 

44 

75569 

24431 

84092 

15908 

08523 

91477 

16 

4 

19 

45 

9.75587 

10.24413 

9.84119 

10.15881 

10.08531 

9.91469 

15 

41 

4 

46 

75605 

24395 

84146 

15854 

08540 

91460 

14 

56 

8 

47 

75624 

24376 

84173 

15827 

08549 

91451 

13 

52 

12 

48 

75612 

24358 

84200 

15800 

08558 

91442 

12 

48 

16 

49 

75660 

24340 

84227 

15773 

08567 

91433 

11 

44 

20 

50 

9.75678 

10.24322 

9.84254 

10.15746 

10.0S575 

9.91425 

10 

40 

24 

51 

75696 

24304 

84280 

15720 

08584 

91416 

9 

36 

28 

52 

75714 

24286 

84307 

15693 

08593 

91407 

8 

32 

32 

53 

75733 

24267 

84334 

15666 

. 08602 

91398 

7 

28 

36 

54 

75751 

24249 

84361 

15639 

08611 

91389 

6 

24 

40 

55 

9.75769 

10.24231 

9.84388 

10.15612 

10.08619 

9.91381 

5 

20 

44 

56 

75787 

24213 

84415 

15585 

08628 

91372 

4 

16 

48 

57 

75805 

24195 

84442 

15558 

08637 

91363 

3 

12 

52 

58 

75823 

24177 

84469 

15531 

08646 

91354 

2 

8 

56 

59 

75841 

24159 

84496 

15504 

08655 

91345 

1 

4 

20 

60 

75859 

24141 

84523 

15477 

03664 

91336 

0 

40 

M.S. 

8 U 

M i Cosine. 

124° 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

55° 

M.S. 

3“ 




















198 


Logarithms Trigonometric. 


2 h 

85° 



Logarithms. 


144° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

1 Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

30 

0 

9.75859 

10.24141 

9.84523 

10.15477 

10.08664 

9.91336 

60 

4:0 

4 

1 

75877 

24123 

84550 

15450 

08672 

91328 

59 

56 

8 

2 

75895 

24105 

84576 

15424 

08681 

91319 

58 

52 

12 

3 

75913 

24087 

84603 

15397 

08690 

91310 

57 

48 

16 

4 

75931 

24069 

84630 

15370 

08699 

91301 

56 

44 

20 

5 

9.75949 

10.24051 

9.84657 

10.15343 

10.08708 

9.91292 

55 

40 

24 

6 

75967 

24033 

84684 

15316 

08717 

91283 

54 

36 

28 

7 

75985 

24015 

84711 

15289 

08726 

91274 

53 

32 

32 

8 

76003 

23997 

84738 

15262 

08734 

91266 

52 

28 

'36 

9 

76021 

23979 

84764 

15236 

08743 

91257 

51 

24 

40 

10 

9.76039 

10.23961 

9.84791 

10.15209 

10.08752 

9.91248 

50 

20 

44 

11 

76057 

23943 

84818 

15182 

08761 

91239 

49 

16 

48 

12 

76075 

23925 

84845 

15155 

08770 

91230 

48 

12 

52 

13 

76093 

23907 

84872 

15128 

08779 

91221 

47 

8 

50 

14 

76111 

23889 

84899 

15101 

08788 

91212 

46 

4 

31 

15 

9.76129 

10.23871 

9.84925 

10.15075 

10.08797 

9.91203 

45 

39 

4 

16 

76146 

23854 

84952 

15048 

08806 • 

91194 

44 

56 

8 

17 

76164 

23836 

84979 

15021 

08815 

91185 

43 

52 

12 

18 

76182 

23818 

85006 

14994 

08824 

91176 

42 

48 

16 

19 

76200 

23800 

85033 

14967 

08833 

91167- 

41 

44 

20 

20 

9.76218 

10.23782 

9.85059 

10.14941 

10.08842 

9.91158 

40 

40 

24 

21 

76236 

23764 

85086 

14914 

08851 

91149 

39 

36 

28 

22 

76253 

23747 

85113 

14887 

08859 

91141 

38 

32 

32 

23 

76271 

23729 

85140 

14860 

08868 

91132 

37 

28 

36 

24 

76289 

23711 

85166 

14834 

08877 

91123 

36 

24 

40 

25 

9.76307 

10.23693 

9.85193 

10.14807 

10.08886 

9.91114 

35 

20 

44 

26 

76324 

23676 

85220 

14780 

08895 

91105 

34 

16 

48 

27 

76342 

23658 

85247 

14753 

08904 

91096 

33 

12 

52 

28 

76360 

23640 

85273 

14727 

08913 

91037 

32 

8 

50 

29 

76378 

23622 

85300 

14700 

08922 

91078 

31 

4 

33 

30 

9.76395 

10.23605 

9.85327 

10.14673 

10.08931 

9.91069 

30 

38 

4 

31 

76413 

23587 

85354 

14646 

08940 

91060 

29 

56 

8 

32 

76431 

23569 

85380 

14620 

08949 

91051 

28 

52 

12 

33 

76448 

23552 

85407 

14593 

08958 

91042 

27 

48 

16 

34 

76466 

23534 

85434 

14566 

08967 

91033 

26 

44 

20 

35 

9.76484 

10.23516 

9.85460 

10.14540 

10.08977 

9.91023 

25 

40 

24 

36 

76501 

23499 

854S7 

14513 

08986 

91014 

24 

36 

28 

37 

76519 

23481 

85514 

14486 

08995 

91005 

23 

32 

32 

38 

76537 

23463 

85540 

14460 

09004 

90998 

22 

28 

36 

39 

76554 

23446 

85567 

14433 

09013 

90987 

21 

24 

40 

40 

9.76572 

10.23428 

9.85594 

10.14406 

10.09022 

9.90978 

20 

20 

44 

41 

76590 

23410 

85620 

14380 

09031 

90969 

19 

16 

48 

42 

76607 

23393 

85647 

14353 

09040 

90960 

18 

12 

52 

43 

76625 

23375 

85674 

14326 

09049 

90951 

17 

8 

56 

44 

76642 

23358 

85700 

14300 

09058 

90942 

16 

4 

33 

45 

9.76660 

10.23340 

9.85727 

10.14273 

10.09067 

9.90933 

15 

37 

4 

46 

76677 

23323 

85754 

14246 

09076 

90924 

14 

56 

8 

47 

76695 

23305 

85780 

14220 

09085 

90915 

13 

52 

12 

48 

76712 

23288 

85807 

14193 

09094 

90906 

12 

48 

16 

49 

76730 

23270 

85834 

14166 

09104 

90896 

11 

44 

20 

50 

9.76747 

10.23253 

9.85860 

10.14140 

10.09113 

9.90887 

10 

40 

24 

51 

76765 

23235 

85887 

14113 

09122 

90878 

9 

36 

28 

52 

76782 

23218 

85913 

14087 

09131 

90869 

8 

32 

32 

53 

76800 

23200 

85940 

14060 

09140 

90860 

7 

28 

36 

54 

76817 

23183 

85967 

14033 

09149 

90851 

6 

24 

40 

55 

9.76835 

10.23165 

9.85993 

10.14007 

10.09158 

9.90842 

5 

20 

44 

56 

76852 

23148 

86020 

139S0 

09168 

90832 

4 

16 

48 

57 

76870 

23130 

86046 

13954 

09177 

90823 

3 

12 

52 

58 

76887 

23113 

86073 

13927 

09186 

90814 

2 

8 

fit) 

59 

76904 

23096 

86100 

13900 

09195 

90805 

1 

4 

34 

60 

76922 

23078 

86126 

13874 

09204 

90796 

0 

36 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

8 h 

125° • 





54° 

3 h 






















Logarithms Trigonometric. 199 


2 h 

36° 



Logarithms. 


143° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

| M 

M.S. 

24: 

0 

9.76922 

10.2307S 

9.86126 

10.13874 

10.09204 

9.90796 

60 

36 

4 

1 

76939 

23061 

86153 

13847 

09213 

90787 

59 

56 

8 

2 

76957 

23043 

86179 

13821 

09223 

90777 

58 

52 

12 

3 

76974 

23026 

86206 

13794 

09232 

90768 

57 

48 

16 

4 

76991 

23009 

86232 

13768 

09241 

90759 

56 

44 

20 

5 

9.77009 

10.22991 

9.86259 

10.13741 

10.09250 

9.90750 

55 

40 

24 

6 

77026 

22974 

86285 

13715 

09259 

90741 

54 

36 

28 

7 

77043 

22957 

86312 

13688 

09269 

90731 

53 

32 

32 

8 

77061 

22939 

86338 

13662 

09278 

90722 

52 

28 

36 

9 

77078 

22922 

86365 

13635 

09287 

90713 

51 

24 

40 

10 

9.77095 

10.22905 

9.86392 

10.13608 

10.09296 

9.90704 

50 

20 

44* 

11 

77112 

22888 

86418 

13582 

09306 

90694 

49 

16 

48 

12 

77130 

22870 

86445 

13555 

09315 

90685 

48 

12 

62 

13 

77147 

22853 

86471 

13529 

09324 

90676 

47 

8 

66 

14 

77164 

22836 

86498 

13502 

09333 

90667 

46 

4 

25 

15 

9.77181 

10.22819 

9.86524 

10.13476 

10.09343 

9.90657 

45 

35 

4 

16 

77199 

22801 

86551 

13449 

09352 

90648 

44 

56 

.8 

17 

77216 

22784 

86577 

13423 

09361 

90639 

43 

52 

12 

18 

77233 

22767 

86603 

13397 

09370 

90630 

42 

48 

16 

19 

77250 

22750 

86630 

13370 

09380 

90620 

41 

44 

20 

20 

9.77268 

10.22732 

9.86656 

10.13344 

10.09389 

9.90611 

40 

40 

24 

21 

77285 

22715 

86683 

13317 

09398 

90602 

39 

36 

28 

22 

77302 

22698 

86709 

13291 

09408 

90592 

38 

32 

32 

23 

77319 

22681 

86736 

13264 

09417 

90583 

37 

28 

36 

24 

77336 

22664 

86762 

13238 

09426 

90574 

36 

24 

40 

25 

9.77353 

10.22647 

9.86789 

10.13211 

10.09435 

9.90565 

35 

20 

44 

26 

77370 

22630 

86815 

13185 

09445 

90555 

34 

16 

48 

27 

77387 

22613 

86842 

13158 

09454 

90546 

33 

12 

62 

28 

77405 

22595 

86868 

13132 

09463 

90537 

32 

8 

56 

29 

77422 

22578 

86894 

13106 

09473 

90527 

31 

4 

26 

30 

9.77439 

10.22561 

9.86921 

10.13079 

10.09482 

9.90518 

30 

34 

4 

31 

77456 

22544 

86947 

13053 

09491 

90509 

29 

56 

8 

32 

77473 

22527 

86974 

13026 

09501 

90499 

28 

52 

12 

33 

77490 

22510 

87000 

13000 

09510 

90490 

27 

48 

16 

34 

77507 

22493 

87027 

12973 

09520 

90480 

26 

44 

20 

35 

9.77524 

10.22476 

9.87053 

10.12947 

10.09529 

9.90471 

25 

40 

24 

36 

77541 

22459 

87079 

12921 

09538 

90462 

24 

36 

28 

37 

77558 

22442 

87106 

12894 

09548 

90452 

23 

32 

32 

38 

77575 

22425 

87132 

12868 

09557 

90443 

22 

28 

36 

39 

77592 

22408 

87158 

12842 

09566 

90434 

21 

24 

40 

40 

9.77609 

10.22391 

9.87185 

10.12815 

10.09576 

9.90424 

20 

20 

44 

41 

77626 

22374 

87211 

12789 

095S5 

90415 

19 

16 

48 

42 

77643 

22357 

87238 

12762 

09595 

90405 

18 

12 

52 

43 

77660 

22340 

87264 

12736 

09604 

90396 

17 

8 

56 

44 

77677 

22323 

87290 

12710 

09614 

90386 

16 

4 

27 

45 

9.77694 

10.22306 

9.87317 

10.12683 

10.09623 

9.90377 

15 

33 

4 

46 

77711 

22289 

87343 

12657 

09632 

90368 

14 

56 

8 

47 

77728 

22272 

87369 

12631 

09642' 

90358 

13 

52 

12 

48 

77744 

22256 

87396 

12604 

09651 

90349 

12 

48 

16 

49 

77761 

22239 

87422 

12578 

09661 

90339 

11 

44 

20 

50 

9.77778 

10.22222 

9.87448 

10.12552 

10.09670 

9.90330 

10 

40 

24 

51 

77795 

22205 

87475 

12525 

09680 

90320 

9 

36 

28 

52 

77812 

22188 

87501 

12499 

09689 

90311 

8 

32 

32 

53 

77829 

22171 

87527 

12473 

09699 

90301 

7 

28 

36 

54 

77846 

22154 

87554 

12446 

09708 

90292 

6 

24 

40 

55 • 

9.77862 

10.22138 

9.87580 

10.12420 

10.09718 

9.90282 

5 

20 

44 

56 

77879 

22121 

87606 

12394 

09727 

90273 

4 

16 

48 

57 

77896 

22104 

87633 

12367 

09737 

90263 

3 

12 

52 

58 

77913 

22087 

87659 

12341 

09746 

90254 

2 

8 

66 

59 

77930 

22070 

87685 

12315 

09756 

90244 

1 

4 

28 

60 

77946 

22054 

87711 

12289 

09765 

90235 

0 

32 

M.S. 

8 b 

M 

126 c 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

53° 

M.S. 

3 b 


















£00 Logarithms Trigonometric. 


2 h 

37 0 



Logarithms. 


142° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

as 

0 

9.77946 

10.22054 

9.87711 

10.12289 

10.09765 

9.90235 

60 

32 

4 

1 

77963 

22037 

87738 

12262 

09775 

90225 

59 

56 

8 

2 

77980 

22020 

87764 

12236 

09784 

90216 

58 

52 

12 

3 

77997 

22003 

87790 

12210 

09794 

90206 

57 

48 

16 

4 

78013 

21987 

87817 

12183 

09803 

90197 

56 

44 

20 

5 

9.78030 

10.21970 

9.87843 

10.12157 

10.09813 

9.90187 

55 

40 

24 

6 

78047 

21953 

87869 

12131 

09822 

90178 

54 

36 

28 

7 

78063 

21937 

87895 

12105 

09832 

9016S 

53 

32 

32 

8 

78080 

21920 

87922 

12078 

09841 

90159 

52 

28 

36 

9 

78097 

21903 

87948 

12052 

09851 

90149 

51 

24 

40 

10 

9.78113 

10.21887 

9.87974 

10.12026 

10.09861 

9.90139 

50 

20 

44 

11 

78130 

21870 

88000 

12000 

09870 

90130 

49 

J6 

48 

12 

78147 

21853 

88027 

11973 

09880 

90120 

48 

12 

52 

13 

78163 

21837 

88053 

11947 

09889 

90111 

47 

8 

56 

14 

78180 

21820 

88079 

11921 

09899 

90101 

46 

4 

39 

15 

9.78197 

10.21803 

9.88105 

10.11895 

10.09909 

9.90091 

45 

31 

4 

16 

78213 

21787 

88131 

11869 

09918 

90082 

44 

56 

8 

17 

78230 

21770 

88158 

11842 

09928 

90072 

43 

52 

12 

18 

78246 

21754 

88184 

11816 

09937 

90063 

42 

48 

16 

19 

78263 

21737 

88210 

11790 

09947 

90053 

41 

44 

20 

20 

9.78280 

10.21720 

9.88236 

10.11764 

10.09957 

9.90043 

40 

40 

24 

21 

78296 

21704 

88262 

11738 

09966 

90034 

39 

36 

28 

22 

78313 

21687 

88289 

11711 

09976 

90024 

38 

32 

32 

23 

78329 

21671 

88315 

11685 

09986 

90014 

37 

28 

36 

24 

78346 

21654 

8S341 

11659 

09995 

90005 

36 

24 

40 

25 

9.78362 

10.21638 

9.88367 

10.11633 

10.10005 

9.89995 

35 

20 

44 

26 

78379 

21621 

88393 

11607 

10015 

89985 

34 

16 

48 

27 

78395 

21605 

88420 

11580 

10024 

89976 

33 

12 

52 

28 

78412 

21588 

88446 

11554 

10034 

89966 

32 

8 

56 

29 

78428 

21572 

88472 

11528 

10044 

89956 

31 

4 

30 

30 

9.78445 

10.21555 

9.88498 

10.11502 

10.10053 

9.89947 

30 

30 

4 

31 

78461 

21539 

8S524 

11476 

10063 

89937 

29 

56 

8 

32 

78478 

21522 

88550 

11450 

10073 

89927 

28 

52 

12 

33 

78494 

21506 

88577 

11423 

10082 

89918 

27 

48 

16 

34 

78510 

21490 

88603 

11397 

10092 

89908 

26 

44 

20 

35 

9.78527 

10.21473 

9.88629 

10.11371 

10.10102 

9.89898 

25 

40 

24 

36 

78543 

21457 

88655 

11345 

10112 

89888 

24 

36 

28 

37 

78560 

21440 

88681 

11319 

10121 

89879 

23 

32 

32 

38 

78576 

21424 

88707 

11293 

10131 

89869 

22 

28 

36 

39 

78592 

21408 

88733 

11267 

10141 

89859 

21 

24 

40 

40 

9.78609 

10.21391 

9.88759 

10.11241 

10.10151 

9.89849 

20 

20 

44 

41 

78625 

21375 

88780 

11214 

10160 

89840 

19 

16 

48 

42 

78642 

21358 

88812 

11188 

10170 

89830 

18 

12 

52 

43 

78658 

21342 

88838 

11162 

10180 

89820 

17 

8 

56 

44 

78674 

21326 

88864 

11136 

10090 

89810 

16 

4 

31 

45 

9.78691 

10.21309 

9.88890 

10.11110 

10.10199 

9.89801 

15 

29 

4 

46 

78707 

21293 

88916 

11084 

10209 

89791 

14 

50 

8 

47 

78723 

« 21277 

88942 

11058 

10219 

89781 

13 

52 

12 

48 

78739 

21261 

88968 

11032 

10229 

89771 

12 

18 

16 

49 

78756 

21244 

88994 

11006 

10239 

89761 

11 

44 

20 

50 

9.78772 

10.21228 

9.89020 

10.10980 

10.10248 

9.89752 

10 

40 

24 

51 

78788 

21212 

89046 

10954 

10258 

89742 

9 

36 

28 

52 

78805 

21195 

89073 

10927 

10268 

89732 

8 

32 

32 

53 

78821 

21179 

89099 

10901 

10278 

89722 

7 

28 

36 

54 

78837 

21163 

89125 

10875 

10288 

89712 

6 

24 

40 

55 

9.78853 

10.21147 

9.89151 

10.10849 

10.10298 

9.89702 

5 

20 

44 

56 

78869 

21131 

89177 

10823 

10307 

89693 

4 

16 

48 

57 

78886 

21114 

89203 

10797 

10317 

89683 

3 

12 

52 

58 

78902 

21098 

89229 

10771 

10327 

89673 

2 

8 

56 

59 

78918 

21082 

89255 

10745 

10337 

89663 

1 

4 

33 

60 

78934 

21066 

89281 

10719 

10347 

89653 

0 

28 

M. S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

S h 

127° 





52° 

3 h 






















Logarithms Trigonometric. 


201 


2 h 

38° 



Logarithms. 


141° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

32 

0 

9.78934 

10.21066 

9.89281 

10.10719 

10.10347 

9.89653 

60 

2§ 

4 

1 

78950 

21050 

89307 

’ 10693 

10357 

89643 

59 

56 

8 

2 

78967 

” 21033 

89333 

10667 

10367 

89633 

58 

52 

12 

3 

78983 

21017 

89359 

10641 

10376 

89624 

57 

48 

16 

4 

78999 

21001 

89385 

10615 

10386 

89614 

56 

44 

20 

5 

9.79015 

10.20985 

9.89411 

10.10589 

10.10396 

9.89604 

55 

40 

24 

6 

79031 

20969 

89437 

10563 

10406 

89594 

54 

36 

28 

7 

79047 

20953 

89463 

10537 

10416 

89584 

53 

32 

32 

8 

79063 

20937 

89489 

10511 

10426 

89574 

52 

28 

36 

9 

79079 

20921 

89515 

10485 

10436 

89564 

51 

24 

40 

10 

9.79095 

10.20905 

9.89541 

10.10459 

10.10446 

9.89554 

50 

20 

44 

11 

79111 

20889 

89567 

10433 

10456 

89544 

49 

16 

48 

12 

79128 

20872 

89593 

10407 

10466 

89534 

48 

12 

52 

13 

79144 

20856 

89619 

10381 

10476 

89524 

47 

8 

56 

14 

79160 

20840 

89645 

10355 

104S6 

89514 

46 

4 

33 

15 

9.79176 

10.20S24 

9.89671 

10.10329 

10.10496 

9.89504 

45 

27 

4 

16 

79192 

26808 

89697 

10303 

10505 

89495 

44 

56 

8 

17 

79208 

20792 

89723 

10277 

10515 

89485 

43 

52 

12 

18 

79224 

20776 

89749 

10251 

10525 

89475 

42 

48 

16 

19 

79240 

20760 

89775 

10225 

10535 

89465 

41 

44 

20 

20 

9.79256 

10.29744 

9.89801 

10.10199 

10.10545 

9.89455 

40 

40 

24 

21 

79272 

20728 

•89S27 

10173 

10555 

89445 

39 

36 

28 

22 

79288 

20712 

89853 

10147 

10565 

89435 

38 

32 

32 

23 

79304 

20696 

89879 

10121 

10575 

89425 

37 

28 

36 

24 

79319 

20681 

89905 

10095 

10585 

89415 

36 

24 

40 

25 

9.79335 

10.20665 

9.89931 

10.10069 

10.10595 

9.89405 

35 

20 

44 

26 

79351 

20649 

89957 

10043 

10605 

89395 

34 

16 

48 

27 

79367 

20633 

89983 

10017 

10645 

893S5 

33 

12 

52 

28 

79383 

20617 

90009 

09991 

10625 

89375 

32 

8 

56 

29 

79399 

20601 

90035 

09965 

10636 

89364 

31* 

4 

34 

30 

9.79415 

10.20585 

9.90061 

10.09939 

10.10646 

9.89354 

30 

26 

4 

31 

79431 

20569 

90086 

09914 

10656 

89344 

29 

56 

8 

32 

79447 

20553 

90112 

09888 

10666 

89334 

28 

52 

12 

33 

79463 

20537 

90138 

09862 

10676 

89324 

27 

48 

16 

34 

79478 

20522 

90164 

09836 

106S6 

89314 

26 

44 

20 

35 

9.79494 

10.20506 

9.90190 

10.09810 

10.10696 

9.89304 

25 

40 

24 

36 

79510 

20490 

90216 

09784 

10706 

89294 

24 

36 

28 

37 

79526 

20474 

90242 

0975$ 

10716 

89284 

23 

32 

32 

38 

79542 

20458 

90268 

09732 

10726 

89274 

22 

28 

36 

39 

79558 

20442 

90294 

09706 

10736 

89264 

21 

24 

40 

40 

9.79573 

10.20427 

9.90320 

10.09680 

10.10746 

9.89254 

20 

20 

44 

41 

79589 

20411 

90346 

09654 

10756 

89244 

19 

16 

48 

42 

79605 

20395 

90371 

09629 

10767 

89233 

18 

12 

52 

43 

79621 

20379 

90397 

09603 

10777 

89223 

17 

8 

56 

44 

79636 

20364 

90423 

09577 

107S7 

89213 

16 

4 

35 

45 

9.79652 

10.20348 

9.90449 

10.09551 

10.10797 

9 89203 

15 

25 

4 

46 

79668 

20332 

90475 

09525 

10807 

89193 

14 

56 

8 

47 

79684 

20316 

90501 

09499 

10817 

89183 

13 

52 

12 

48 

79699 

20301 

90527 

09473 

10827 

89173 

12 

48 

16 

49 

79715 

20285 

90553 

09447 

10838 

89162 

11 

44 

20 

50 

9.79731 

10.20269 

9.90578 

10.09422 

10.10848 

9.89152 

10 

40 

24 

51 

79746 

20254 

90604 

09396 

10858 

89142 

9 

36 

28 

52 

79762 

20238 

90630 

09370 

10868 

89132 

8 

32 

32 

53 

79778 

20222 

90656 

09344 

10878 

89122 

7 

28 

36 

54 

79793 

20207 

90682 

09318 

108 S 8 

89112 

6 

24 

40 

55 

9.79809 

10.20191 

9.90708 

10.09292 

10.10899 

9.89101 

5 

20 

44 

56 

79825 

20175 

90734 

09266 

10909 

89091 

4 

16 

48 

57 

79840 

*20160 

90759 

09241 

10919 

89081 

3 

12 

52 

58 

79856 

20144 

90785 

09215 

10929 

89071 

2 

8 

56 

59 

79872 

' 20128 

90811 

09189 

10940 

89060 

1 

4 

36 

60 

79887 

20113 

90837 

09163 

10950 

89050 

0 

24 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sme. 

M 

M. S. 

S h 1128 

D 






51° 























202 Logarithms Trigonometric. 


2 b 

39 c 



Logarithms. 


140 c 

9 h 

M.S. 

M 

Siue. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

36 

0 

9.79887 

10.20113 

9.90837 

10.09163 

10.10950 

9.89050 

60 

3L 

4 

l 

79903 

20097 

90863' 

09137 

1096JD 

89040 

59 

56 

8 

2 

79918 

20082 

90889 

09111 

10970 

89030 

58 

52 

12 

3 

79934 

20066 

90914 

09086 

10980 

89020 

57 

48 

16 

4 

79950 

2(1050 

90940 

09060 

10991 

89009 

56 

44 

20 

5 

9.79965 

10..20035 

9.90906 

10.09034 

10.11001 

9.88999 

55 

40 

24 

6 

79981 

20019 

90992 

09008 

11011 

88989 

54 

36 

28 

7 

79996 

20004 

91018 

08982 

11022 

88978 

53 

32 

32 

8 

80012 

19988 

91043 

08957 

11032 

88968 

52 

28 

36 

9 

80027 

19973 

91069 

0S931 

11042 

88958 

51 

24 

40 

10 

9.80043 

10.19957 

9.91095 

10.08905 

10.11052 

9.88948 

50 

20 

44 

11 

80058 

19942 

91121 

08879 

11063 

88937 

49 

10 

48 

12 

80074 

19926 

91147 

08853 

11073 

88927 

48 

12 

52 

13 

80089 

19911 

91172 

08828 

11083 

88917 

47 

8 

56 

14 

80105 

19895 

91198 

08802 

11094 

88906 

46 

4 

37 

15 

9.80120 

10.19880 

9.91224 

10.08776 

10.11104 

9.88896 

45 

33 

4 

16 

80136 

19864 

91250 

08750 

11114 

88886 

44 

56 

8 

17 

80151 

19849 

91276 

08724 

11125 

88875 

43 

52 

12 

18 

80166 

19834 

91301 

08699 

11135 

88865 

42 

48 

16 

19 

80182 

19818 

91327 

08673 

11145 

88855 

41 

44 

20 

20 

9.80197 

10.19803 

9.91353 

10.08647 

10.11156 

9.88844 

40 

40 

24 

21 

80213 

19787 

91379 

08621 

11166 

88834 

39 

36 

28 

22 

80228 

19772 

91404 

0S596 

11176 

88824 

3S 

32 

32 

23 

80244 

19756 

91430 

08570 

11187 

88813 

37 

28 

36 

24 

80259 

19741 

91456 

08544 

11197 

88803 

36 

24 

40 

25 

9.80274 

10.19726 

9.91482 ' 

10.0S518 

10.11207 

9.88793 

35 

20 

44 

26 

80290 

19710 

91507 

0S493 

11218 

8S782 

34 

16 

48 

27 

80305 

19695 

91533 

08467 

11228 

88772 

33 

12 

52 

28 

80320 

19680 

91559 

08441 

11239 

88761 

32 

8 

56 

29 

80336 

19664 

91585 

08415 

11249 

88751 

31 

4 

38 

30 

9.80351 

10.19649 

9.91610 

10.08390 

10.11259 

9.88741 

30 

33 

4 

31 

80366 

19634 

91636 

08364 

11270 

88730 

29 

56 

8 

32 

80382 

19618 

91662 

08338 

11280 

88720 

28 

52 

12 

33 

80397 

19603 

91688 

08312 

11291 

88709 

27 

48 

16 

34 

80412 

19588 

91713 

08287 

11301 

88699 

26 

44 

20 

35 

9.80428 

10.19572 

9.91739 

10.08261 

10.11312 

9.88688 

25 

40 

24 

36 

80443 

19557 

91765 

08235 

11322 

88678 

24 

36 

28 

37 

80458 

19542 

91791 

08209 

11332 

88668 

23 

32 

32 

38 

80473 

19527 

91816 

08184 

11343 

88657 

22 

28 

36 

39 

80489 

19511 

91842 

08158 

11353 

88647 

21 

24 

40 

40 

9.80504 

10.19496 

9.91868 

10.08132 

10 11364 

9.88636 

20 

20 

44 

41 

80519 

19481 

91893 

08107 

11374 

88626 

19 

16 

48 

42 

80534 

19466 

91919 

08081 

11385 

88615 

18 

12 

62 

43 

80550 

19450 

91945 

08055 

11395 

88605 

17 

8 

66 

44 

80565 

19435 

91971 

08029 

11406 

88594 

16 

4 

39 

45 

9.80580 

10.19420 

9.91996 

10.08004 

10.11416 

9 88584 

15 

31 

4 

46 

80595 

19405 

92022 

07978 

11427 

88573 

14 

56 

8 

47 

80610 

19390 

92048 

07952 

11437 

88563 

13 

52 

12 

48 

80625 

19375 

92073 

07927 

11448 

88552 

12 

48 

16 

49 

80641 

19359 

92099 

07901 

11458 

88542 

11 

44 

20 

50 

9.80656 

10.19344 

9.92125 

10.07875 

10.11469 

9.88531 

10 

40 

24 

51 

80671 

19329 

92150 

07850 

11479 

88521 

9 

36 

28 

52 

80686 

19314 

92176 

07824 

11490 

88510 

8 

32 

32 

53 

80701 

19299 

92202 

07798 

11501 

88499 

7 

28 

36 

54 

80716 

19284 

92227 

07773 

11511 

88489 

6 

24 

40 

55 

9.80731 

1019269 

9.92253 

10.07747 

1011522 

9.88478 

5 

20 

44 

56 

80746 

19254 

92279 

07721 

11532 

8S468 

4 

16 

48 

57 

80762 

19238 

92304 

07696 

11543 

88457 

3 

12 

52 

58 

80777 

19223 

92330 

07670 

11553 

88447 

2 

8 

56 

59 

80792 

19208 

92356 

07644 

11564 

88436 

1 

4 

4LO 

60 

80807 

19193 

92381 

07619 

11575 

88425 

0 

30 

M.S. 

8 h 

M 

129 

Cosine. 

D 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

50° 

M. S. 


















2 h 

o 

0 

M.S. 

M 

LO 

0 

4 

1 

8 

2 

12 

3 

16 

4 

20 

5 

24 

6 

28 

7 

32 

8 

36 

9 

40 

10 

44 

11 

48 

12 

52 

13 

56 

14 

41 

15 

4 

16 

8 

17 

12 

18 

16 

19 

20 

20 

24 

21 

28 

22 

32 

23 

36 

24 

40 

25 

44 

26 

48 

27 

52 

28 

56 

29 

4a 

30 

4 

31 

8 

32 

12 

33 

16 

34 

20 

35 

24 

36 

28 

37 

32 

38 

36 

39 

40 

40 

44 

41 

48 

42 

52 

43 

56 

44 

43 

45 

4 

46 

8 

47 

12 

48 

16 

49 

20 

50 

24 

51 

28 

52 

32 

53 

36 

54 

40 

55 

44 

56 

48 

57 

52 

58 

56 

59 

44 

60 

M. S. 

M 

8 h I 

130 


Sine. 
9.80807 
80822 
80837 
80852 
80867 
9.80882 
80897 
80912 
80927 
80942 
9.80957 
80972 
80987 
81002 
81017 
9.81032 
81047 
81061 
81076 
81091 
9.81106 
81121 
81136 
81151 
81166 
9.81180 
81195 
81210 
81225 
81240 
9.81254 
81269 
81284 
81299 
81314 
9.81328 
81343 
8135S 
81372 
81387 
9.81402 
81417 
81431 
81446 
81461 
9.81475 
81490 
81505 
81519 
81534 
9.81549 
81563 
81578 
81592 
81607 
9.81622 
81636 
81651 
81665 
81680 
81694 
Cosine. I 


Tangent. 

9.92381 

92407 

92433 

92458 

92484 

9.92510 

92535 

92561 

92587 

92612 

9.92638 

92663 

92689 

92715 

92740 

9.92766 

92792 

92817 

92843 

92868 

9.92894 

92920 

92945 

92971 

92996 

9.93022 

93048 

93073 

93099 

93124 

9.93150 

93175 

93201 

93227 

93252 

9.93278 

93303 

93329 

93354 

93380 

9.93406 

93431 

93457 

93482 

93508 

9.93533 

93559 

93584 

93610 

93636 

9.93661 

93687 

93712 

93738 

93763 

9.93789 

93814 

93840 

93865 

93891 

93916 

Cotangent 


Cotangent. 

10.07619 

07593 

07567 

07542 

07516 

10.07490 

07465 

07439 

07413 

07388 

10.07362 

07337 

07311 

07285 

07260 

10.07234 

07208 

07183 

07157 

07132 

10.07106 

07080 

07055 

07029 

07004 

10.06978 

06952 

06927 

06901 

06876 

10.06850 

06825 

06799 

06773 

06748 

10.06722 

06697 

06671 

06646 

06620 

10.06594 

06569 

06543 

06518 

06492 

10.06467 

06441 

06416 

06390 

06364 

10.06339 

06313 

06288 

06262 

06237 

10.06211 

06186 

06160 

06135 

06109 

06084 

Tangent. 


Secant. 

10.11575 

11585 

11596 

11606 

11617 

10.11628 

11638 

11649 

11660 

11670 

10.11681 

11692 

11702 

11713 

11724 

10.11734 

11745 

11756 

11766 

11777 

10.11788 

11799 

11809 

11820 

11831 

10.11842 

11852 

11863 

11874 

11885 

10.11895 

11906 

11917 

11928 

11939 

10.11949 

11960 

11971 

11982 

11993 

10.12004 

12015 

12025 

12036 

12047 

10.12058 

12069 

12080 

12091 

12102 

10.12113 

12123 

12134 

12145 

12156 

10.12167 

12178 

12189 

12200 

12211 

12222 

Cosecant. 



203 

59° 

A 

M 

M.S. 

60 

30 

59 

56 

58 

52 

57 

48 

56 

44 

55 

40 

54 

36 

53 

32 

52 

28 

51 

24 

50 

20 

49 

16 

48 

12 

47 

8 

46 

4 

45 

10 

44 

56 

43 

52 

42 

48 

41 

44 

40 

40 

39 

36 

38 

32 

37 

28 

36 

24 

35 

20 

34 

16 

33 

12 

32 

8 

3L 

4 

30 

IS 

29 

56 

28 

52 

27 

48 

26 

44 

25 

40 

24 

36 

23 

32 

22 

28 

21 

24 

20 

20 

19 

16 

18 

12 

17 

8 

16 

4 

15 

17 

14 

56 

13 

52 

12 

48 

11 

44 

10 

40 

9 

36 

8 

32 

7 

28 

6 

24 

5 

20 

4 

16 

3 

12 

2 

8 

1 

4 

0 

in' 

M 

M.S. 

9° 

3 11 


Logarithms Trigonometric. 


Logarithms. 


lc 


Cosecant. 
10.19193 
19178 
19163 
19148 
19133 
10.19118 
19103 
19088 
19073 
19058 
10.19043 
19028 
19013 
18998 
18983 
10.18968 
18953 
18939 
18924 
18909 
10.18894 
18S79 
18864 
18849 
18834 
10.18820 
18805 
18790 
18775 
18760 
10.18746 
18731 
18716 
18701 
18686 
10.18672 
18657 
18642 
18628 
18613 
10.18598 
18583 
18569 
18554 
18539 
10.18525 
18510 
18495 
18481 
18466 
10.18451 
1S437 
18422 
18408 
18393 
10.18378 
18364 
18349 
18335 
18320 
18306 
Secant. 


Cosine. 

9.88425 

88415 

88404 

88394 

883S3 

9.88372 

88362 

88351 

88340 

88330 

9.88319 

88308 

88298 

88287 

88276 

9.88266 

88255 

88244 

88234 

88223 

9.88212 

88201 

88191 

88180 

88169 

9.88158 

88148 

88137 

88126 

88115 

9.88105 

88094 

88083 

88072 

88061 

9.88051 

88040 

88029 

88018 

88007 

9.87996 

87985 

87975 

87964 

87953 

9.87942 

87931 

87920 

87909 

87898 

9.87887 

87877 

87866 

87855 

87844 

9.87833 

87822 

87811 

87800 

87789 

87778 

Sine. 
























204 Logarithms Trigonometric. 


2 h 

41° 



Logarithms. 


138° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangeut. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

44 

0 

9.81694 

10.18306 

9.93916 

10.06084 

10.12222 

9.87778 

60 

16 

4 

1 

„ 81709 

18291 

93942 

06058 

12233 

87767 

59 

56 

8 

2 

81723 

18277 

93967 

06033 

12244 

87756 

58 

52 

12 

3 

81738 

18262 

93993 

06007 

12255 

87745 

57 

48 

16 

4 

81752 

18248 

94018 

05982 

12266 

87734 

56 

44 

20 

5 

9.81767 

10.18233 

9.94044 

10.05956 

10.12277 

9.87723 

55 

40 

24 

6 

81781 

18219 

94069 

05931 

12288 

87712 

54 

56 

28 

7 

81796 

18204 

94095 

05905 

12299 

87701 

53 

32 

32 

8 

81810 

18190 

94120 

05880 

12310 

87690 

52 

28 

36 

9 

81825 

18175 * 

94146 

05854 

12321 

87679 

51 

24 

40 

10 

9.81839 

10.18161 

9.94171 

10.05829 

10.12332 

9.87663 

50 

20 

44 

11 

81854 

18146 

94197 

05803 

12343 

87657 

49 

16 

48 

12 

81868 

18132 

94222 

05778 

12354 

87646 

48 

12 

52 

13 

81882 

18118 

94248 

05572 

12365 

87635 

47 

8 

56 

14 

81897 

18103 

94273 

05727 

12376 

87624 

46 

4 

45 

15 

9.81911 

10.18089 

9.94299 

10.05701 

10.12387 

9.876 i 3 

45 

15 

4 

16 

81926 

18074 

94324 

05676 

12399 

87 601 

44 

56 

8 

17 

81940 

18060 

94350 

05650 

12410 

87590 

43 

52 

12 

18 

81955 

18045 

94375 

05625 

12421 

87579 

42 

48 

16 

19 

81969 

18031 

94401 

05599 

12432 

87568 

41 

44 

20 

20 

9.81983 

10.18017 

9.94426 

10.05574 

10.12443 

9.87557 

40 

40 

24 

21 

81998 

18002 

94452 

05548 

12454 

87546 

39 

36 

28 

22 

82012 

17988 

94477 

05523 

12465 

87535 

38 

32 

32 

23 

82026 

17974 

94503 

05497 

12476 

87524 

37 

28 

36 

24 

82041 

17959 

94528 

05472 

12487 

87513 

36 

24 

40 

25 

9.32055 

10.17945 

9.94554 

10.05446 

10.12499 

9.87501 

35 

20 

44 

26 

82069 

17931 

94579 

05421 

12510 

87490 

34. 

16 

48 

27 

82084 

17916 

94604 

05396 

12521 

87479 

33 

12 

52 

28 

82098 

17902 

94630 

05370 

12532 

87468 

32 

8 

56 

29 

82112 

17888 

94655 

05345 

12543 

87457 

31 

4 

46 

30 

9.82126 

10.17874 

9.94681 

10.05319 

10.12554 

9.87446 

30 

14 

4 

31 

82141 

17859 

94706 

05294 

12566 

87434 

29 

56 

8 

32 

82155 

17845 

94732 

05268 

12577 

87423 

28 

52 

12 

33 

82169 

17831 

94757 

05243 

12588 

87412 

27 

48 

16 

34 

82184 

17816. 

94783 

05217 

12599 

87401 

26 

44 

20 

35 

9.82198 

10.17802 

9.94808 

10.05192 

10.12610 

9.87390 

25 

40 

24 

36 

82212 

17788 

94834 

05166 

12622 

87378 

24 

36 

28 

37 

82226 

17774 

94859 

05141 

12633 

87367 

23 

32 

32 

38 

82240 

17760 

94884 

05116 

12644 

87356 

22 

28 

36 

39 

82255 

17745 

94910 

05090 

12655 

87345 

21 

24 

40 

40 

9.82269 

10.17731 

9.94935 

10.05065 

10.12666 

9.87334 

20 

20 

44 

41 

82283 

17717 

94961 

05039 

12678 

87322 

19 

16 

48 

42 

82297 

17703 

94986 

05014 

12689 

87311 

18 

12 

52 

43 

82311 

17689 

95012 

04988 

12700 

87300 

17 

8 

56 

44 

82326 

17674 

95037 

04963 

12712 

87288 

16 

4 

47 

45 

9.82340 

10.17660 

9.95062 

10.04938 

10.12723 

9.87277 

15 

13 

4 

46 

82354 

17646 

95088 

04912 

12734 

87266 

14 

56 

8 

47 

82368 

17632 

95113 

04887 

12745 

87255 

13 

52 

12 

48 

82382 

17618 

95139 

04861 

12757 

87243 

12 

48 

16 

49 

82396 

17604 

95164 

04836 

12768 

87232 

11 

44 

20 

50 

9.82410 

10.17590 

9.95190 

10.04810 

10.12779 

9.87221 

10 

40 

24 

51 

82424 

17576 

95215 

04785 

12791 

87209 

9 

36 

28 

52 

82439 

17561 

95240 

04760 

12802 

87198 

8 

32 

32 

53 

82453 

17547 

95266 

04734 

12813 

87187 

7 

28 

36 

54 

82467 

17533 

95291 

04709 

12825 

87175 

6 

24 

40 

55 

9.82481 

10.17519 

9.95317 

10.04683 

10.12836 

9.87164 

5 

20 

44 

56 

'82495 

17505 

95342 

04658 

12847 

87153 

4 

16 

48 

57 

82509 

17491 

95368 

04632 

12859 

87141 

3 

12 

52 

58 

'82523 

17477 

95393 

04607 

12870 

87130 

2 

8 

56 

59 

82537 

17463 

95418 

04582 

12881 

87119 

1 

4 

48 

60 

82551 

17449 

95444 

04556 

12*93 

87107 

0 

13 

M. S. 
8 h 

M 

131 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

48° 

M.S. 

3* 


















Logarithms Trigonometric. 205 


2 h 

42° 



Logarithms. 


137° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

48 

0 

9.82551 

10.17449 

9.95444 

10.04556 

10.12893 

9.87107 

60 

12 

4 

l 

82565 

17435 

95469 

04531 

12904 

87096 

59 

56 

8 

2 

82579 

17421 

95495 

04505 

12915 

87085 

58 

*52 

12 

3 

82593 

17407 

95520 

04480 

12927 

87073 

57 

48 

16 

4 

82607 

17393 

95545 

04455 

12938 

87062 

56 

4-1 

20 

5 

9.82621 

10.17379 

9.95571 

10.04429 

10.12950 

9.87050 

55 

40 

24 

6 

82635 

17365 

95596 

04404 

12961 

87039 

54 

36 

28 

7 

82649 

17351 

95622 

04378 

12972 

87028 

53 

32 

32 

8 

82663 

17337 

95647 

04353 

12984 

87016 

52 

28 

36 

9 

82677 

17323 

95672 

04328 

12995 

87005 

51 

24 

40 

10 

9.82691 

10.17309 

9.95698 

10.04302 

10.13007 

9.86993 

50 

20 

44 

11 

82705 

17295 

95723 

04277 

13018 

86982 

49 

16 

48 

12 

82719 

17281 

95748 

04252 

13030 

86970 

48 

12 

52 

13 

82733 

17267 

95774 

04226 

13041 

86959 

47 

8 

56 

14 

82747 

17253 

95799 

04201 

13053 

86947 

46 

4 

49 

15 

9.82761 

10.17239 

9.95825 

10.04175 

10.13064 

9.86936 

45 

11 

4 

16 

82775 

17225 

95850 

04150 

13076 

86924 

44 

56 

8 

17 

82788 

17212 

95875 

04125 

13U87 

86913 

43 

52 

12 

18 

82802 

17198 

95901 

04099 

13098 

86902 

42 

48 

16 

19 

82816 

17184 

95926 

04074 

13110 

86890 

41 

44 

20 

20 

9.82830 

10.17170 

9.95952 

10.04048 

10.13121 

9.86879 

40 

40 

24 

21 

82844 

17156 

95977 

04023 

13133 

86S67 

39 

36 

28 

22 

82858 

17142 

96002 

03998 

13145 

86855 

38 

32 

32 

23 

82872 

17128 

96028 

03972 

13156 

86844 

37 

28 

36 

24 

82885 

17115 

96053 

03947 

13168 

86832 

36 

24 

40 

25 

9.82899 

10.17101 

9.96078 

10.03922 

10.13179 

9.8682 L 

35 

20 

44 

26 

82913 

17087 

96104 

03896 

13191 

86809 

34 

16 

48 

27 

82927 

17073 

96129 

03871 

13202 

86798 

33 

12 

52 

28 

82941 

17059 

96155 

03845 

13214 

867S6 

32 

8 

56 

29 

82955 

17045 

96180 

03820 

13225 

86775 

31 

4 

50 

30 

9.82968 

10.17032 

9.96205 

10.03795 

10.13237 

9.86763 

30 

10 

4 

31 

82982 

17018 

96231 

03769 

13248 

86752 

29 

56 

8 

32 

82996 

17004 

96256 

03744 

13260 

S6740 

28 

52 

12 

33 

83010 

16990 

96281 

03719 

13272 

86,'28 

27 

48 

16 

34 

83023 

16977 

96307 

03693 

13283 

86717 

26 

44 

20 

35 

9.83037 

10.16963 

9.96332 

10.03668 

10.13295 

9.86705 

25 

40 

24 

36 

83051 

16949 

96357 

03643 

13306 

86694 

24 

36 

28 

37 

83065 

16935 

96383 

03617 

13318 

86652 

23 

32 

32 

38 

83078 

16922 

96408 

03592 

13330 

86670 

22 

28 

36 

39 

83092 

16908 

96433 

03567 

13341 

86659 

21 

24 

40 

40 

9.83106 

10.16894 

9.96459 

10.03541 

10.13353 

9.86647 

20 

20 

44 

41 

83120 

16880 

96484 

03516 

13365 

86635 

19 

16 

48 

42 

83133 

16S67 

96510 

03490 

13376 

86624 

18 

12 

52 

43 

83147 

16853 

96535 

03465 

13388 

86612 

17 

8 

56 

44 

83161 

16839 

96560 

03440 

13400 

86600 

16 

4 

51 

45 

9.83174 

10.16826 

9.96586 

10.034 L4 

10.13411 

9.86589 

15 

9 

4 

46 

83188 

16812 

96611 

03389 

13423' 

86577 

14 

56 

8 

47 

83202 

16798 

96636 

03304 

13435 

86565 

13 

52 

12 

48 

83215 

16785 

96662 

03338 

13446 

86554 

12 

48 

16 

49 

83229 

16771 

96687 

03313 

13458 

86542 

11 

44 

20 

50 

9.83242 

10.16758 

9.96712 

10.03288 

10.13470 

9.86530 

10 

40 

24 

51 

83256 

16744 

96738 

03202 

13482 

86518 

9 

36 

28 

52 

83270 

16730 

96763 

03237 

13493 

86507 

8 

32 

32 

53 

83283 

16717 

96788 

03212 

13505 

86495 

7 

28 

36 

54 

83297 

16703 

96814 

03186 

13517 

86483 

6 

24 

40 

55 

9.83310 

10.16690 

9.96839 

10.03161 

10.13528 

9.86472 

5 

20 

44 

56 

83324 

16676 

96864 

03136 

13540 

86460 

4 

16 

48 

57 

83338 

16662 

96890 

03110 

13552 

86448 

3 

12 

52 

58 

83351 

16649 

96915 

03085 

13564 

86436 

2 

8 

5*5 

59 

83365 

16635 

96940 

03060 

13575 

86425 

1 

4 

sa 

60 

83378 

16622 

96966 

03034 

13587 

86413 

0 

8 

M. S. 
8 h 

M 

132 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

47° 

M.S. 

3 h 


















208 


Logarithms Trigonometric. 


Oh ! 

AJ 

43° 



Logarithms. 


136° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S 

53 

0 

9.83378 

10.16622 

9.96966 

10.03034 

10.13587 

9.86413 

60 

8 

4 

1 

83392 

10608 

96991 

03009 

13599 

86401 

59 

56 

8 

2 

83405 

10595 

97016 

02984 

13611 

86389 

58 

52 

12 

3 

83419 

16581 

97042 

02958 

13623 

86377 

57 

48 

16 

4 

83432 

16568 

97067 

02933 

13634 

86366 

56 

44 

20 

5 

9.83446 

10.16554 

9.97092 

10.02908 

10.13646 

9.86354 

55 

40 

24 

6 

83459 

16541 

97118 

02882 

1365S 

86342 

54 

36 

28 

7 

83473 

16527 

97143 

02857 

13670 

8G330 

53 

32 

32 

8 

83486 

16514 

97168 

02832 

13682 

86318 

52 

28 

36 

9 

83500 

16500 

97193 

02807 

13694 

86306 

51 

24 

40 

10 

9.83513 

10.16487 

9.97219 

10.02781 

10.13705 

9.86295 

50 

20 

44 

11 

83527 

16473 

97244 

02756. 

13717 

86283 

49 

16 

48 

12 

83540 

16400 

97269 

02734 

13729 

86271 

48 

12 

52 

13 

83554 

16446 

97295 

02705 

13741 

86259 

47 

8 

56 

14 

83567 

16433 

97320 

02680 

13763 

86247 

46 

4 

53 

15 

9.83581 

10.10419 

9.97345 

10.02655 

10.13765 

9.86235 

45 

7 

4 

16 

83594 

16406 

97371 

02629 

13777 

86223 

44 

56 

8 

17 

83608 

16392 

97396 

02604 

13789 

86211 

43 

52 

12 

18 

83621 

16379 

97421 

02579 

13800 

86200 

42 

48 

16 

19 

83634 

16366 

97447 

02553 

13812 

86188 

41 

44 

20 

20 

9.83648 

10.16352 

9.97472 

10.02528 

10.13824 

9.86176 

40 

40 

24 

21 

83661 

16339 

97497 

02503 

13836 

86164 

39 

36 

28 

22 

83674 

16326 

97523 

02477 

13848 

86152 

38 

32 

32 

23 

83688 

16312 

97548 

02452 

13860 

86140 

37 

28 

36 

24 

83701 

16299 

97573 

02427 

13872 

86128 

36 

24 

40 

25 

9.83715 

10.10285 

9.97598 

10.02402 

10.13884 

9.86116 

35 

20 

44 

26 

83728 

16272 

97624 

02376 

13896 

86104 

34 

16 

48 

27 

83741 

16259 

97649 

02351 

13908 

86092 

33 

12 

52 

28 

83755 

16245 

97674 

02326 

13920 

86080 

32 

8 

56 

29 

83768 

16232 

97700 

02300 

13932 

86068 

31 

4 

54 

30 

9.83781 

10.16219 

9.97725 

10.02275 

10.13944 

9.86056 

30 

6 

4 

31 

83795 

16205 

97750 

02250 

13956 

86044 

29 

56 

8 

32 

83808 

16192 

97776 

02224 

13968 

86032 

28 

52 

12 

33 

83821 

16179 

97801 

02199 

13980 

86020 

27 

48 

16 

34 

83834 

16106 

97826 

02174 

13992 

86008 

26 

44 

20 

35 

9.83848 

10.16152 

9.97851 

10.02149 

10.14004 

9.85996 

25 

40 

24 

36 

83861 

16139 

97877 

02123 

14016 

85984 

24 

36 

28 

37 

83874 

10126 

97902 

02098 

14028 

85972 

23 

32 

32 

38 

83887 

10113 

97927 

02073 

14040 

85960 

22 

28 

36 

39 

83901 

16099 

97953 

02047 

14052 

85948 

21 

24 

40 

40 

9.83914 

10.16086 

9.97978 

10.02022 

10.14064 

9.85936 

20 

20 

44 

41 

83927 

16073 

98003 

01997 

14076 

85924 

19 

16 

48 

42 

83940 

16060 

98029 

01971 

14088 

85912 

18 

12 

52 

43 

83954 

16046 

98054 

01946 

14100 

85900 

17 

8 

56 

44 

83967 

16033 

98079 

01921 

14112 

85888 

16 

4 

55 

45 

9.83980 

10.16020 

9.98104 

10.01896 

10.14124 

9.85876 

15 

5 

4 

46 

83993 

16007 

98130 

01870 

14136 

85864 

14 

56 

8 

47 

84006 

15994 

98155 

01845 

14149 

85851 

13 

52 

12 

48 

84020 

15980 

98180 

01820 

14161 

85839 

12 

48 

16 

49 

84033 

15967 

98206 

01794 

14173 

85827 

11 

44 

20 

50 

9.84046 

10.15954 

9.98231 

10.01769 

10.14185 

9.85815 

10 

40 

24 

51 

84059 

15941 

98256 

01744 

14197 

85803 

9 

36 

28 

52 

84072 

15928 

98281 

01719 

14209 

85791 

8 

32 

32 

53 

84085 

15915 

98307 

01693 

14221 

85779 

7 

28 

36 

54 

84098 

15902 

98332 

01668 

14234 

85766 

6 

24 

40 

55 

9.84112 

10.15888 

9.98357 

10.01643 

10.14246 

9.85754 

5 

20 

44 

56 

S4125 

15875 

98383 

01617 

14258 

85742 

4 

16 

48 

57 

84138 

15862 

98408 

01592 

14270 

85730 

3 

12 

52 

58 

84151 

15849 

98433 

01567 

14282 

85718 

2 

8 

5b 

59 

84164 

15836 

98458 

01542 

14294 

85706 

1 

4 

5G 

60 

84177 

15823 

98484 

01516 

14307 

85693 

0 

4 

M.S. 

8 b 

M 

133 

1 Cosine, 
o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

46° 

M.S. 

3 h 





















Logarithms Trigonometric. 


207 


2“ 

44° 



Logarithms. 


135° 

9 h 

M .S. 

M 

Sine.. 

Cosecant. 

Tangent. 

Cotangeut. 

Secant. 

Cosine. 

M 

M. S. 

56 

0 

9.84177 

10.15823 

9.98484 

10.01516 

10.14307 

9.85693 

60 

4: 

4 

1 

84190 

15810 

98509 

• 01491 

14319 

85681 

59 

56 

8 

2 

84203 

15797 

98534 

01166 

14331 

85669 

58 

52 

12 

3 

84216 

15784 

98560 

01440 

14343 

85657 

57 

48 

16 

4 

84229 

15771 

98585 

01415 

14355 

85645 

56 

44 

20 

5 

9.84242 

10.1575S 

9.98610 

10.01390 

10.14368 

9.85632 

55 

40 

24 

6 

84255 

15745 

98635 

01365 

14380 

85620 

54 

36 

28 

7 

84269 

15731 

98661 

01339 

14392 

85608 

53 

32 

32 

8 

84282 

15718 

986S6 

01314 

14404 

85596 

52 

28 

36 

9 

• 84295 

15705 

98711 

01289 

14417 

85583 

51 

24 

40 

10 

9.84308 

10.15692 

9.98737 

10.01263 

10.14429 

9.85571 

50 

20 

44 

11 

84321 

15679 

98762 

01238 

14441 

S5559 

49 

16 

48 

12 

84334 

15666 

98787 

01213 

14453 

85547 

48 

12 

52 

13 

84347 

15653 

98812 

01188 

14466 

85534 

47 

8 

56 

14 

84360 

15640 

98838 

01162 

14478 

85522 

46 

4 

51 

15 

9.84373 

10.15627 

9.98863 

10.01137 

10.14490 

9.85510 

45 

3 

4 

16 

84385 

15615 

98388 

01112 

14503 

85497 

44 

56 

8 

17 

84398 

15602 

98913 

01087 

14515 

85485 

43 

52 

12 

18 

84411 

15589 

98939 

01061 

14527 

85473 

42 

48 

10 

19 

84424 

15576 

98964 

01036 

14540 

85460 

41 

44 

20 

20 

9.84437 

10.15563 

9.98989 

10.01011 

10.14552 

9.85448 

40 

40 

24 

21 

84450 

15550 

99015 

00985 

14564 

85436 

39 

36 

28 

22 

84463 

15537 

99040 

00960 

14577 

85423 

38 

32 

32 

23 

84476 

15524 

99065 

00935 

14589 

85411 

37 

28 

36 

24 

84489 

15511 

99090 

00910 

14601 

85399 

36 

24 

40 

25 

9.84502 

10.15498 

9.99116 

10.00884 

10.14614 

9.85386 

35 

20 

44 

26 

84515 

15485 

99141 

00859 

14626 

85374 

34 

16 

48 

27 

84528 

15472 

99166 

00834 

14639 

85361 

33 

12 

52 

28 

84540 

15460 

99191 

00309 

14651 

85349 

32 

8 

56 

29 

84553 

15447 

99217 

00783 

14663 

85337 

31 

4 

58 

30 

9.84566 

10.15434 

9.99242 

10.00758 

10.14676 

9.85324 

30 

a 

4 

31 

84579 

15421 

99267 

00733 

14688 

85312 

29 

56 

8 

32 

84592 

15408 

99293 

00707 

14701 

85299 

28 

52 

12 

33 

84605 

15395 

99318 

00682 

14713 

85287 

27 

48 

16 

34 

84618 

15382 

99343 

00657 

14726 

85274 

26 

44 

20 

35 

9.84630 

10.15370 

9.99368 

10.00632 

10.14738 

9.85262 

25 

40 

24 

36 

84643 

15357 

99394 

00606 

14750 

85250 

24 

36 

28 

37 

84656 

15344 

99419 

00581 

14763 

85237 

23 

32 

32 

38 

84669 

15331 

99444 

00556 

14775 

85225 

22 

28 

36 

39 

84682 

15318 

99469 

00531 

14788 

85212 

21 

24 

4o 

40 

9.84694 

10.15366 

9.99495 

10.00505 

10.14800 

9.85200 

20 

20 

44 

41 

84707 

15293 

99520 

00480 

14813 

851S7 

19 

16 

48 

42 

84720 

15280 

99545 

00455 

14825 

85175 

18 

12 

52 

43 

84733 

15267 

99570 

00430 

14838 

85162 

17 

8 

56 

44 

84745 

15255 

99596 

00404 

14850 

85150 

16 

4 

5‘J 

45 

9.84758 

10.15242 

9.99621 

10.00379 

10.14863 

9 85137 

15 

1 

4 

46 

84771 

15229 

99646 

00354 

14875 

85125 

14 

56 

8 

47 

84784 

16216 

99672 

00328 

14888 

85112 

13 

52 

12 

48 

84796 

15204 

99697 

00303 

14900 

85100 

12 

48 

16 

49 

84809 

15191 

99722 

00278 

14913 

85087 

11 

44 

20 

50 

9.84822 

10.15178 

9.99747 

10.00253 

10.14926 

9.85074 

10 

40 

24 

51 

84835 

15165 

99773 

00227 

14938 

85062 

9 

36 

28 

52 

84847 

15153 

99798 

00202 

14951 

85049 

8 

32 

32 

53 

84860 

15140 

99823 

00177 

14963 

85037 

7 

28 

3(3 

54 

84873 

15127 

99S48 

00152 

14976 

85024 

6 

24 

40 

55 

9.84885 

10.15115 

9.99874 

10.00126 

1014988 

9.85012 

5 

20 

44 

56 

84898 

15102 

99899 

00101 

15001 

84999 

4 

16 

48 

57 

84911 

15089 

99924 

00076 

15014 

84986 

3 

12 

52 

58 

84923 

15077 

99949 

00051 

15026 

84974 

2 

8 

56 

59 

84936 

15064 

99975 

00025 

15039 

84961 

1 

4 

60 

60 

84949 

15051 

10.00060 

OOOOO 

15051 

84949 

0 

O 

M. S. 
8 h 

M 

134 

Cosine. 

o 

Secant. 

Cotangent. 

Tangent. 

Coseoant. 

Sine. 

M 

45° 

.U. S. 

3 h 





















208 


Explanation op the Tables. 


EXPLANATION OF THE TABLES. 

Tho outer columns in the trigonometrical tables contain the angle in time of 
hours, minutes and seconds, corresponding to the same angle in degrees and min¬ 
utes ih the next columns. The hour is noted at the top and bottom, the minutes 
in black, and the seconds in ordinary figures. 

To find tlie Logarithm and Natural Line for Seconds exceeding 

Minutes of a Degree. 

Example 1. Find the logarithm for sin. 38° 47' 55". 

’log. sin. 3S° 48'= 9.79699' 


diff. 15. 


... flog. sin. 3S° 48'= 9.796991 
From table, | „ 38 o p-p = 9.79034 } 1 

Correction, 15 X 55 : 60 = +14 n early. 

The required log. sin. 38° 47' 55" = 9.79698 
In practice, the difference is subtracted direct from the tables. 

Example 2. Find the natural cos. 43° 29' 19". 

From table, cos. 43° 29' = 0.72557 
Correction, 20 X 19 : 60 = —6 n early. 

The required cos. 43° 29' 19" = 0.72551 

The correction is added when the function is increasing, and subtracted when 
decreasing. 

To find the Angle corresponding to a given Logarithm or Nat¬ 
ural Line. 

Example 3. Log. sin. = 9.56429. Required the angle. 

' ‘ )g. sin. 21° 31' = 9.56440 \ diff g9 

21° 30' = 9.564081 

The angle required, “ “ 21° 30' 29"= 9.56429 J * 

Correction, 21 X 60 : 32 = 29 seconds nearly. 

Example 4. Cosine = 0.35254. Required the angle. 

From table, { co *‘ 6y ~ 22 ' = °- 35239 j diff. 27. 

*- “ 69° 21' = 0.35266 < 

The required angle, “ 69° 21' 27" = 0.35254 J ^ 2 ‘ 

Correction, 12 X 60 : 27 = 27 seconds, nearly. 

Conversion of Minutes and Seconds into Decimals of a Degree 

or of an Hour. 


From table. 


ft 


Decimal 

M. 

Decimal. 

s. 

Decimal. 

S. 

Decimal. 

s. 

Decimal. 

.350000 

41 

.683333 

] 

.000277 

21 

.005833 

41 

.011388 

.366666 

42 

.700000 

2 

.000555 

22 

.006111 

42 

.011666 

.383333 

43 

.716666 

3 

.000833 

23 

.006388 

43 

.011944 

.400000 

44 

.733333 

4 

.001111 

24 

.006666 

44 

.012222 

.416666 

45 

.750000 

5 

.001388 

25 

.006914 

45 

.012500 

.433333 

46 

.766666 

6 

.001666 

26 

.007222 

46 

.012777 

.450000 

47 

.783333 

7 

.001914 

27 

.007500 

47 

.013055 

.466666 

48 

.800000 

8 

.002222 

28 

.007777 

48 

.013333 

.483333 

49 

.816666 

9 

.002500 

29 

.008055 

49 

.013611 

.500000 

50 

833333 

10 

.002777 

30 

.008333 

50 

.013888 

.516000 

51 

.850000 

11 

.003055 

31 

.008611 

51 

.011166 

.533333 

52 

.866666 

12 

.003333 

32 

.008888 

52 

.014141 

.550000 

53 

.883333 

13 

.003611 

33 

.009166 

53 

.014722 

.566666 

54 

.900000 

14 

.003888 

34 

.009114 

54 

.015000 

.583333 

55 

.916666 

15 

.004166 

35 

.009722 

55 

.015277 

.600000 

56 

.933333 

16 

.004441 

36 

.010000 

56 

.015555 

.616606 

57 

.950000 

17 

.004722 

37 

.010277 

57 

.015833 

.633333 

58 

.966666 

18 

.005000 

38 

.010555 

58 

.016111 

.650000 

59 

.983333 

19 

.005277 

39 

.010833 

59 

.016388 

.666666 

00 

l.COOOOO 

20 

.005555 

40 

.011111 

GO 

.016666 


M. 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 


Decimal. 

.016666 

.033333 

.050000 

.066666 

.083333 

.100000 

.116666 

.133333 

.150000 

.106606 

.183333 

.200000 

.216666 

.233333 

.250000 

266666 

.283333 

.300000 

.316666 

.333333 


M. 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 




















Natural Lines 


200 


0 ° 

Natural Trig 

onometrical 

Functions. 

179° 

11* 

M. 

Sine. 1 

Yrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin. 

Cosine. 

M. 

M.S. 

0 

.00000 

1.0000 

Infinite 

.00000 

Infinite 

1.0000 

.00000 

1.0000 

60 

GO 

l 

. 0029 

.99971 

3437.7 

. 0029 

3437.7 

.0000 

. oooo 

.0000 

59 

56 

2 

. 0058 

. 9942 

1718.9 

. 0058 

1718.9 

.0000 

. oooo 

.0000 

58 

52 

3 

. 0087 

. 9913 

1145.9 

. 0087 

1145.9 

.0000 

. oooo 

.0000 

57 

48 

4 

. 0116 

. 9884 

859.44 

. 0116 

859.44 

.0000 

. oooo 

.0000 

56 

44 

5 

.00145 

.99854 

6S7.55 

.00145 

687.55 

1.0000 

.00000 

1.0000 

55 

40 

e 

. 0174 

. 9825 

572.96 

. 0174 

572.96 

.oooo 

. oooo 

.0000 

54 

36 

7 

. 020 !: 

. 9796 

491.11 

. 0204 

491.11 

.oooo 

. oooo 

.0000 

53 

32 

8 

. 0233 

. 9767 

429.72 

. 0233 

429.72 

.0000 

. oooo 

.0000 

52 

28 

9 

. 0262 

. 9738 

3 4.97 

. 0262 

381.97 

.0000 

. oooo 

.0000 

51 

24 

10 

.00291 

.99709 

343.71 

.00291 

343.77 

1.0000 

.00000 

.99999 

50 

20 

11 

. 0320 

. 9680 

312.52 

. 0320 

312.52 

.0000 

. oooo 

. 9999 

49 

16 

12 

. 0349 

. 9651 

286.48 

. 0349 

286 48 

.0000 

. 0001 

. 9999 

48 

12 

13 

. 0378 

. 9622 

64.44 

. 0378 

64.44 

.0000 

. 0001 

. 9999 

47 

8 

14 

. 0407 

. 9593 

45.55 

. 0407 

45 55 

.0000 

. 0001 

. 9939 

46 

,4 

16 

.00436 

.99564 

229.18 

.00436 

229.18 

1.0000 

.00001 

.99999 

45 

59 

16 

. 0465 

. 9534 

14.86 

. 0465 

14.86 

.0000 

. 0001 

. 9999 

44 

56 

17 

. 0494 

. 9505 

02.22 

. 0434 

02.22 

.0000 

. 0001 

. 9999 

43 

52 

18 

. 0524 

. 9476 

190.99 

. 0524 

190.98 

.0000 

. 0001 

. 9999 

42 

48 

19 

. 0553 

. 9447 

180.93 

. 0553 

180.93 

.0000 

. 0001 

. 9998 

41 

44 

2(J 

.00582 

.99418 

171.89 

.00582 

171.88 

1.0000 

.00002 

.99998 

40 

40 

21 

. 0611 

. 9389 

63.70 

. 0611 

63.70 

.0000 

. 0002 

. 9998 

39 

36 

22 

. 0640 

. 9360 

56.26 

. 0640 

56.26 

.0000 

. 0002 

. 9998 

38 

32 

23 

. 0669 

. 9331 

49.47 

. 0669 

49.46 

.0000 

. 0002 

. 9998 

37 

28 

24 

. 0698 

. 9302 

43.24 

. 0698 

43.24 

.0000 

. 0002 

. 9997 

36 

24 

25 

.00727 

.99273 

137.51 

.00727 

137.51 

1.0000 

.00003 

.99997 

35 

20 

26 

. 0766 

. 9244 

32.22 

. 0756 

32.22 

.0000 

. 0003 

. 9997 

34 

16 

27 

. 0785 

. 9215 

27.32 

. 0785 

27.32 

.0000 

. 0003 

. 9997 

33 

12 

28 

. 0814 

. 9185 

22.78 

. 0814 

22.77 

.0000 

. 0003 

. 9997 

32 

8 

29 

. 0813 

. 9156 

18.54 

. 0844 

18 54 

.0000 

. 0003 

. 9996 

31 

4 

30 

.00873 

.99127 

114.59 

.00873 

114.59 

1.0000 

.00004 

.99996 

30 

58 

31 

. 0902 

. 9098 

10.90 

. Q9o2 

10.89 

.0000 

. 0004 

. 9996 

29 

56 

32 

. 0931 

. 9069 

07.43 

. 0931 

07.43 

.0000 

. 0004 

. 9996 

28 

52 

33 

. 0960 

. 9040 

04.17 

. 0960 

04.17 

.0000 

. 0005 

. 9995 

27 

48 

31 

. 0989 

. 9011 

01.11 

. 0989 

01.11 

.0000 

. 0005 

. 9995 

26 

44 

35 

.01018 

.98982 

98.223 

.01018 

98.218 

1.0000 

.00005 

.99995 

25 

40 

36 

. 1047 

. 8953 

5.495 

. 1047 

5.489 

.0000 

. 0005 

. 9994 

21 

36 

37 

. 1076 

. 8924 

2.914 

. 1076 

2.908 

.0000 

. 0006 

. 9994 

23 

32 

38 

. 1105 

. 8895 

0.469 

. 1105 

0.463 

.0001 

. 0006 

. 9994 

22 

28 

39 

. 1134 

. 8865 

88.149 

. 1134 

88.143 

.0001 

. 0006 

. 9993 

21 

21 

40 

.01163 

.98836 

85.946 

.01164 

85.940 

1.0001 

.00007 

.79993 

20 

20 

41 

. 1193 

. 8807 

3.849 

. 1193 

3.843 

.0001 

. 0007 

. 9993 

19 

16 

42 

. 1222 

. 8778 

1.853 

. 1222 

1.847 

.0001 

. 0007 

. 9992 

18 

12 

43 

. 1251 

. 8749 

79.950 

. 1251 

79.913 

.0001 

. 0008 

. 9992 

17 

8 

44 

. 1280 

. 8720 

78.133 

. 1280 

78.126 

.01 01 

. 0008 

. 9992 

16 

4 

45 

.01309 

.98691 

76.396 

.01309 

76.890 

1.0001 

.00008 

.99991 

15 

57 

46 

. 1338 

. 8662 

4.736 

. 1338 

4.729 

.0001 

. 0009 

. 9991 

14 

56 

47 

. 1367 

. 8633 

3.146 

. 1367 

3.13 * 

.0001 

. 0009 

. 9991 

13 

52 

48 

. 1396 

. 8604 

1.622 

. 1396 

1.615 

.0001 

. 0010 

. 9990 

12 

48 

49 

. 1425 

. 8575 

0.160 

. 1425 

0.153 

.0001 

. 0010 

. 9990 

11 

44 

50 

.01454 

.98546 

68.757 

.01454 

68.750 

1.0001 

.00010 

.99989 

10 

40 

51 

. 1483 

. 8516 

7.409 

.-1484 

7.402 

.0001 

. 0011 

. 9989 

9 

36 

52 

. 1512 

. 8487 

6.113 

. 1513 

6.105 

.0001 

. 0011 

. 9988 

8 

32 

53 

. 1542 

. 8458 

4.866 

. 1542 

4.858 

.0001 

. 0012 

. 9988 

7 

28 

64 

. 1571 

. 8429 

3.664 

. 1571 

3.657 

.0001 

. 0012 

. 9988 

6 

21 

55 

.01600 

.98400 

62.507 

.01600 

62.499 

1.0001 

.00013 

.99987 

5 

20 

56 

. 1629 

. 8371 

1.391 

. 1629 

1.383 

.0001 

. 0013 

. 9987 

4 

16 

01 

. 1658 

. 8342 

0.314 

. 1658 

0.306 

.0001 

. 0014 

. 99S7 

3 

12 

58 

. 16S7 

. 8313 

59.274 

. 16S7 

59.2 ,; 6 

.0001 

. 0014 

. 9986 

2 

8 

59 

. 1716 

. 8284 

8.270 

. 1716 

8 261 

.0001 

. 0015 

. 9985 

1 

4 

60 

. 1745 

. 8255 

7.299 

. 1745 

7.290 

.0001 

. 0015 

. 9985 

0 

50 

M. 

90° 

Cosine. 

Vrs.Sin. 

. Secante. 

Cotang. Tangent. 

Natural. 

Cosec'ut 

Vrs.Cos. 

Sine. 

M. 

89° 

M.S. 

5 h 


14 
























Natural Lines. 


210 


(> h 

1 ° 

JVatni’al Trigonometrical Functions. 

178° 

ll h 

M.S 

M 

Sine. 

Yrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Sceante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

4 

0 

.01745 

.98255 

57.299 

.01745 

57.290 

1.0001 

.00015 

.99985 

60 

56 

4 

1 

. 1774 

. 8226 

56.359 

. 1775 

56.350 

.0001 

. 0016 

. 9984 

59 

56 

8 

2 

. 1803 

. 8196 

55.450 

. 1804 

55.441 

.0002 

. 0016 

. 9984 

58 

62 

12 

3 

. 1832 

. 8167 

54.570 

. 1833 

54.561 

.0002 

. 0017 

. 9983 

67 

48 

1(5 

4 

. 1861 

. 8138 

53.718 

. 1862 

53 708 

.0002 

. 0017 

. 9983 

50 

44 

20 

5 

.01891 

.98109 

52.891 

.01891 

52.882 

1.0002 

.00018 

.99982 

55 

40 

24 

6 

. 1920 

. 8080 

2.090 

. 1920 

2.081 

.0002 

. 0018 

. 9981 

54 

36 

28 

7 

. 1949 

. 8051 

1.313 

. 1949 

1.303 

.0002 

. 0019 

. 9981 

53 

32 

32 

8 

. 1978 

. 8022 

0.558 

. 1978 

0.548 

.0002 

. 0019 

. 9980 

52 

28 

3(5 

9 

. 2007 

. 7993 

49.826 

. 2007 

49.816 

.0002 

. 0020 

. 9980 

51 

24 

40 

10 

.02036 

.97964 

49.114 

.(12036 

40.104 

1.0002 

.00021 

.99979 

50 

20 

45 

11 

. 2065 

. 7935 

8.422 

. 2066 

8.412 

.0002 

. 0021 

. 9979 

49 

16 

48 

12 

. 2094 

. 7906 

7.750 

. 2095 

7.739 

.0002 

. 0022 

. 9978 

48 

12 

52 

13 

. 2123 

. 7877 

7.096 

. 2124 

7.085 

.0002 

. 0022 

. 9977 

47 

8 

56 

14 

. 2152 

. 7847 

6.460 

. 2153 

6.449 

.0002 

. 0023 

. 9977 

46 

4 

5 

15 

.02181 

.97818 

45.840 

.021S2 

45.829 

1.0002 

.00024 

.99976 

45 

55 

4 

16 

. 2210 

. 7789 

5.237 

. 2211 

5.226 

.0002 

. 0024 

. 9975 

44 

56 

s 

17 

. 2240 

. 7760 

4.650 

. 2240 

4.638 

.0002 

. 0025 

. 9975 

43 

52 

12 

18 

. 2269 

. 7731 

4.077 

. 2269 

4.066 

.0002 

. 0026 

. 9974 

42 

48 

16 

19 

. 2298 

. 7702 

3.520 

. 2298 

3.508 

.0003 

. 0026 

. 9974 

41 

44 

20 

20 

.02327 

.97673 

42.976 

.02327 

42.964 

1.0003 

.00027 

.99973 

40 

40 

24 

21 

. 2356 

. 7644 

2.445 

. 2357 

2.433 

.0003 

. 0028 

. 9972 

39 

36 

28 

22 

. 2385 

. 7615 

1.928 

. 2386 

1.916 

.0003 

. 0028 

. 9971 

38 

32 

32 

23 

. 2414 

. 7586 

1.423 

. 2415 

1.410 

.0003 

. 0029 

. 9971 

37 

28 

36 

24 

. 2443 

.. 7557 

0.930 

. 2444 

0.917 

.0003 

. 0030 

. 9970 

36 

24 

40 

25 

.02472 

.97528 

40.448 

.02473 

40.436 

1.0003 

.00030 

.99969 

35 

20 

44 

26 

. 2501 

. 7499 

39.978 

. 2502 

39.965 

.0003 

. 0031 

. 9969 

34 

16 

4S 

27 

. 2530 

. 7469 

9.518 

. 2531 

9.506 

.0003 

. 0032 

. 9968 

33 

12 

52 

28 

. 2559 

. 7440 

9.069 

. 2560 

9.057 

.0003 

. 0033 

. 9967 

32 

8 

56 

29 

. 2589 

. 7411 

8.631 

. 2589 

8.618 

.0003 

. 0033 

. 9966 

31 

4 

6 

30 

.02618 

.97382 

38.201 

.02618 

38.188 

1.0003 

.00034 

.99966 

30 

54 

4 

31 

. 2647 

. 7353 

7.782 

. 2648 

7.769 

.0003 

. 0035 

. 9965 

29 

56 

8 

32 

. 2676 

. 7324 

7.371 

. 2677 

7.358 

.0003 

. 0036 

. 9964 

28 

52 

12 

33 

. 2705 

. 7295 

6.969 

. 2706 

6.956 

.0004 

. 0036 

. 9963 

27 

48 

10 

34 

. 2734 

. 7266 

6.576 

. 2735 

6.663 

.0004 

. 0037 

. 9963 

26 

44 

20 

35 

.02763 

.97237 

36.191 

.02764 

36.177 

1.0004 

.00038 

.99962 

25 

40 

24 

36 

. 2792 

. 7208 

5.814 

. 2793 

5.800 

.0004 

. 0039 

. 9961 

24 

36 

28 

37 

. 2S21 

. 7179 

5.445 

. 2822 

5.431 

.0004 

. 0040 

. 9960 

23 

32 

32 

38 

. 2850 

. 7150 

5.084 

. 2851 

5.069 

.0004 

. 0041 

. 9959 

22 

28 

36 

39 

. 2879 

. 7121 

4.729 

. 2880 

4.715 

.0004 

. 0041 

. 9958 

21 

24 

40 

40 

.02908 

.97091 

34.382 

.02910 

34.368 

1.0004 

.00042 

.99958 

20 

20 

44 

41 

. 2937 

. 7062 

4.042 

. 2939 

4.027 

.0004 

. 0043 

. 9957 

19 

16 

48 

42 

. 2967 

. 7033 

3.708 

. 2968 

3.693 

.0004 

. 0044 

. 9956 

18 

12 

52 

43 

. 2996 

. 7004 

3.381 

. 2997 

3.366 

.0004 

. 0045 

. 9955 

17 

8 

56 

44 

. 3025 

. 6975 

3.060 

. 3026 

3.045 

.0004 

. 0046 

. 9954 

16 

4 

7 

45 

.03054 

.96946 

32.745 

.03055 

32.730 

1.0005 

.00046 

.99953 

15 

53 

4 

46 

. 3083 

. 9692 

2.437 

. 3084 

2.421 

.0005 

. 0047 

. 9952 

14 

66 

8 

47 

. 3112 

. 0S88 

2.134 

. 3113 

2.118 

.0005 

. 0048 

. 9951 

13 

52 

12 

48 

. 3141 

. 6859 

1.836 

. 3143 

1.820 

.0005 

. 0049 

. 9951 

12 

48 

16 

49 

. 3170 

. 6830 

1.544 

. 3172 

1.528 

.0005 

. 0050 

. 9950 

11 

44 

20 

50 

.03199 

.96801 

31.257 

.03201 

31.241 

1.0005 

.00051 

.99949 

10 

40 

24 

51 

. 3228 

. 6772 

0.978 

. 3230 

0.960 

.0005 

. 0052 

. 9948 

9 

36 

28 

52 

. 3257 

. 6743 

0.699 

. 3259 

0.683 

.0005 

. 0053 

. 9917 

8 

32 

32 

53 

. 3286 

. 6713 

0.428 

. 3288 

0.411 

.0006 

. 0054 

. 9946 

7 

28 

36 

54 

. 3315 

. 6684 

0.161 

. 3317 

0.145 

.0005 

. 0055 

. 9945 

6 

24 

40 

55 

.03344 

.96655 

29.899 

.03346 

29.882 

1.0005 

.00056 

.99944 

5 

20 

44 

56 

. 3374 

. 6626 

9.641 

. 3375 

9.624 

.0006 

. 0057 

. 9943 

4 

16 

48 

57 

. 3403 

. 6597 

9.388 

. 3405 

9.371 

.0006 

. 0058 

. 9942 

3 

12 

52 

58 

. 3432 

• 6568 

9.139 

. 3434 

9.122 

.0006 

. 0059 

. 9941 

2 

8 

56 

59 

. 3461 

. 6539 

8.894 

. 3463 

8.877 

.0006 

. 0060 

. 9940 

1 

4 

8 

60 

. 3490 

. 6510 

8.654 

. 3492 

8.636 

.0006 

. 0061 

. 9939 

0 

52 

MS. 

M 

Cosine. 

Vrs.Sin- 

Secante. 

Cotang. Tangent. 

Cosee’nt 

Vrs.Cos 

Sine. 

M 

M.S. 

6 b 

91° 




Natural. 




88° 

5 h 



























Natural Lines. ' 211 


o h 

2° 

Natural Trigonometrical 

Functions. 

177° 

ll h 

M.S. 

* 

Sine. 

Yrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Yrs. Sin 

Cosine. 

M 

M.S. 

8 


.03490 

.96510 

2S 654 

.03492 

28 636 

1.0006 

.00061 

.99939 

60 

5 a 

4 

l 

. 3519 

. 6481 

8.417 

. 3521 

8.399 

.0006 

. 0062 

. 9938 

59 

56 

8 

2 

3 

. 3548 

. 6452 

8.184 

. 3550 

8.166 

.0006 

. 0063 

. 9937 

58 

52 

12 

. 3577 

. 6423 

7.955 

. 3579 

7.937 

.0006 

. 0064 

. 9936 

57 

48 

16 

4 

. 3606 

•. 6394 

7.730 

. 3608 

7.712 

.0006 

. 0065 

. 99:15 

56 

44 

20 

5 

.03635 

.96365 

27.508 

.03638 

27.490 

1.0007 

.00066 

.99934 

55 

40 

24 

6 

. 366 1 

. 6336 

7.290 

. 3667 

7.271 

.0007 

. 0067 

. 9933 

54 

36 

2b> 

7 

. 3693 

. 6306 

7.075 

. 3696 

7.056 

.0007 

. 0068 

. 9932 

53 

32 

32 

8 

. 3722 

. 6277 

6.S64 

. 3725 

6.845 

.0007 

. 0069 

. 9931 

52 

2S 

36 

9 

. 3751 

. 6248 

6 655 

. 3754 

6.637 

.0007 

. 0070 

. 9930 

51 

24 

40 

10 

.03781 

.96219 

26.450 

.03783 

26.432 

1.0007 

.00071 

.99928 

50 

20 

41 

11 

. 3810 

. 6190 

6 249 

. 3812 

6.230 

.0007 

. 0073 

. 9927 

49 

16 

48 

12 

. 3839 

. 6161 

6.050 

. 3812 

6.031 

.0007 

. 0074 

. 9926 

48 

12 

52 

13 

. 3868 

. 6132 

5.854 

. 3871 

5.835 

.0007 

. 0075 

. 9925 

47 

8 

56 

9 

14 

. 3897 

. 6103 

5.661 

. 3900 

5.642 

.0008 

. 0076 

. 9924 

46 

4 

15 

.03926 

.96074 

25.471 

.03929 

25.452 

1.0008 

.00077 

.99923 

45 

51 

4 

16 

. 3955 

. 6045 

5.284 

. 3958 

5.264 

.0008 

. 0078 

. 9922 

44 

56 

8 

17 

. 3984 

. 6016 

5.100 

. 3987 

5.080 

.0008 

. 0079 

. 9921 

43 

52 

12 

18 

. 4013 

. 5987 

4.918 

. 4016 

4.898 

.0008 

. 0080 

. 9919 

42 

48 

16 

19 

. 4042 

. 5958 

4.739 

. 4045 

4.718 

.0008 

. 0082 

. 9918 

41 

44 

20 

20 

.04071 

.95929 

24.562 

.04075 

24.542 

1.0008 

.00083 

.99917 

40 

40 

24 

21 

. 4100 

. 5900 

4.388 

. 4104 

4.367 

.0008 

. 0084 

. 9916 

39 

36 

28 

22 

. 4129 

. 5870 

4.216 

. 4133 

4.196 

.0008 

. 0085 

. 9915 

38 

32 

32 

36 

23 

. 4158 

. 5841 

4.047 

. 4162 

4.026 

.0009 

. 0086 

. 9913 

37 

28 

24 

. 4187 

. 5812 

3 880 

. 4191 

3.S59 

.0009 

. 0088 

. 9912 

36 

24 

40 

25 

.04217 

.95783 

23.716 

.04220 

23.694 

1.0009 

.00089 

.99911 

35 

20 

44 

26 

. 4246 

. 5754 

3.753 

. 4249 

3.532 

.0009 

. 0090 

. 9910 

34 

16 

48 

27 

. 4275 

. 5725 

3.393 

. 4279 

3.372 

.0009 

. 0091 

. 9908 

33 

12 

52 

28 

. 4304 

. 5696 

3.235 

. 4308 

3.214 

.0009 

. 0093 

. 9907 

32 

8 

56 

29 

. 4333 

. 5667 

3.079 

. 4337 

3.058 

.0009 

. 0094 

. 9906 

31 

4 

30 

30 

.04362 

.95638 

22.925 

.04366 

22.904 

1.0009 

.00095 

.99905 

30 

50 

4 

31 

. 4391 

. 5609 

2.774 

. 4395 

2.752 

.0010 

. 0096 

. 9903 

29 

56 

8 

32 

. 4420 

. 5580 

2.624 

. 4424 

2.602 

.0010 

. CM )98 

. 9902 

28 

62 

12 

33 

. 4449 

. 5551 

2.476 

. 4453 

2.454 

.0010 

. 0099 

. 9901 

27 

48 

16 

34 

. 4478 

. 5522 

2.330 

. 4483 

2.308 

.0010 

. 0100 

. 9900 

26 

44 

20 

35 

.04507 

.95493 

22.1S6 

.04512 

22.164 

1.0010 

.00102 

.99898 

25 

40 

24 

36 

. 4536 

. 5464 

2.044 

. 4541 

2.022 

.0010 

. 0103 

. 9897 

24 

36 

28 

37 

. 4565 

. 5435 

1.904 

. 4570 

1.881 

.0010 

. 0104 

. 9896 

23 

32 

32 

38 

. 4594 

. 5405 

1.765 

. 4599 

1.742 

.0010 

. 0106 

. 9894 

22 

28 

36 

39 

. 4623 

. 5376 

1.629 

. 4628 

1.606 

.0011 

. 0107 

. 9893 

21 

24 

40 

40 

.04652 

.95347 

21.494 

.04657 

21.470 

1.0011 

.00108 

.99892 

20 

20 

44 

41 

. 4681 

. 5318 

1.360 

. 4687 

1.337 

.0011 

. 0110 

. 9890 

19 

16 

48 

42 

. 4711 

. 5289 

1.228 

. 4716 

1.205 

.0011 

. 0111 

. 9889 

18 

12 

52 

43 

. 4710 

. 5260 

1.098 

. 4745 

1.075 

.0011 

. 0112 

. 9888 

17 

8 

56 

44 

. 4769 

. 5231 

0.970 

. 4774 

0.946 

.own 

. 0114 

. 9886 

16 

4 

11 

45 

.04798 

.95202 

20.843 

.04803 

20.819 

1.0011 

.00115 

.99885 

15 


4 

46 

. 4827 

. 5173 

0.717 

. 4832 

0.693 

.0012 

. 0116 

. 9883 

14 

56 

8 

47 

. 4S56 

. 5144 

0.593 

. 4862 

0.569 

.0012 

. 0118 

. 9882 

13 

52 

12 

48 

. 4885 

. 5115 

0.471 

. 4891 

0.446 

.0012 

. 0119 

. 9881 

12 

48 

16 

49 

. 4914 

. 5086 

0.350 

. 4920 

0.325 

.0012 

. 0121 

. 9879 

11 

44 

20 

50 

.04943 

.95057 

20.200 

.04949 

20.205 

1.0012 

.00122 

.99S78 

10 

40 

24 

51 

. 4972 

. 5028 

0.112 

. 4978 

0.087 

.0012 

. 0124 

. 9876 

9 

36 

28 

52 

. 5001 

. 4999 

19.995 

. 5007 

19.970 

.0012 

. 0125 

. 9875 

8 

32 

32 

53 

. 5030 

. 4970 

9.880 

. 5037 

9.854 

.0013 

. 0127 

. 9873 

7 

28 

36 

54 

. 5059 

. 4941 

9.766 

. 5066 

9.710 

.0013 

. 0128 

. 9872 

6 

24 

40 

55 

.05088 

.94912 

19.653 

.05095 

19.627 

1.0013 

.00129 

.99870 

5 

20 

41 

56 

. 5117 

. 4883 

9.541 

. 5124 

9.515 

.0013 

. 0131 

. 9869 

4 

16 

48 

57 

. 5146 

. 4853 

9.431 

. 5153 

9.405 

.0013 

. 0132 

. 9867 

3 

12 

52 

58 

. 5175 

. 4824 

9.322 

. 5182 

9.296 

.0013 

. 0134 

. 9866 

2 

8 

56 

59 

. 5204 

. 4795 

9.214 

. 5212 

9.188 

.0013 

. 0135 

. 9864 

1 

4 

12 

60 

. 5234 

. 4766 

9.107 

. 5241 

9.081 

.0014 

. 0137 

. 9863 

0 

4:8 

M.S. 

6 h 

. 

M 

92° 

Cosine. 

Yrs.Sin. 

Secante. I 

Cotaug. ‘Tangent. 

Natural. 

Coseo'nt 

Yrs.Cos 

Sine. 

M 

87° 

M.S. 

5 h 




























Natural Lines. 


212 


O h , 3° 

Natural Trigonometrical Functions 

176° 

ll h 

M.S. 

M 

Sine. 

VTs.Cos. SCosec'nte 

! Tang. 

Cotang 

Seeante. JVrs. Sir 

Cosine 

M 

M.S. 

i.-i 

0 

.05234 

.94766 

19.107 

.05241 

19.081 

1.0014 

.00137 

.99863 

60 

48 

4 

1 

. 5263 

. 4737 

9.002 

. 5270 

8.975 

.0014 

j. 0138 

. 9861 

59 

56 

8 

2 

. 5292 

. 4708 

8.897 

. 5299 

8.871 

.0014 

. 0140 

. 9860 

58 

52 

12 

3 

. 5321 

. 4679 

8.794 

. 5328 

8.768 

.0014 

. 0142 

. 9858 

57 

48 

1G 

4 

. 5350 

. 4650 

8.692 

. 5357 

8.665 

.0014 

. 0143 

. 9857 

56 

44 

20 

5 

.05379 

.94621 

18.591 

.05387 

18.564 

1.0014 

.00145 

.99855 

55 

40 

24 

6 

. 5408 

. 4592 

8.491 

. 5416 

8.464 

.0015 

. 0146 

. 9854 

54 

36 

28 

7 

. 5437 

. 4563 

8.393 

. 5445 

8.3.65 

.0015 

. 0148 

. 9852 

53 

32 

32 

8 

. 5466 

. 4534 

8.295 

. 5474 

8.268 

.0015 

. 0149 

. 9850 

52 

28 

36 

9 

. 5495 

. 4505 

8.198 

. 5503 

8.171 

.0015 

. 0151 

. 9S49 

51 

24 

40 

10 

.05524 

.94476 

18.103 

.05532 

18.075 

1.0015 

.00153 

.99847 

50 

20 

41 

11 

. 5553 

. 4447 

8.008 

. 5562 

7.980 

.0015 

. 0154 

. 9846 

49 

16 

48 

12 

. 5582 

. 4418 

7.914 

. 5591 

7.886 

.0016 

. 0156 

. 9844 

48 

12 

52 

13 

. 5611 

. 4389 

7.S21 

. 5620 

7.793 

.0016 

. 0157 

. 9S42 

47 

8 

56 

14 

. 5640 

. 4360 

7.730 

. 5649 

7.701 

.0016 

. 0159 

. 9841 

46 

4 

33 

15 

.05669 

.94331 

17.639 

.05678 

17.610 

1.0016 

.00161 

.9:1839 

45 

47 

4 

16 

. 1 698 

. 4302 

7.549 

. 5707 

7.520 

.0016 

. 0162 

. 9837 

44 

56 

8 

17 

. 5727 

. 4273 

7.460 

. 5737 

7.431 

.0016 

. 0164 

. 9836 

43 

52 

12 

18 

. 5756 

. 4241 

7.372 

. 5766 

7.343 

.0017 

. 0166 

. 9834 

42 

48 

16 

19 

. 5785 

. 4214 

7.285 

. 5795 

7.256 

.0017 

. 0167 

. 9832 

41 

44 

20 

20 

.05814 

.94185 

17.198 

.05824 

17.169 

1.0017 

.00169 

.99831 

40 

40 

24 

21 

. 5843 

. 4156 

7.113 

. 5853 

7.084 

.0017 

. 0171 

. 9829 

39 

36 

28 

22 

. 5872 

. 4127 

7.028 

. 5883 

6.999 

.0017 

. 0172 

. 9827 

38 

32 

32 

23 

. 5902 

. 4098 

6.944 

. 5912 

6.915 

.0017 

. 0174 

. 9826 

37 

28 

36 

24 

. 5931 

. 4069 

6.861 

. 5941 

6.832 

.0018 

. 0176 

. 9824 

36 

24 

40 

25 

.05960 

.94040 

16.779 

.05970 

16.750 

1.0018 

.00178 

.99822 

35 

20 

44 

26 

. 5989 

. 4011 

6.69S 

. 5999 

6.668 

.0018 

. 0179 

. 9820 

34 

16 

48 

27 

. 6018 

. 3982 

6.617 

. 6029 

6.587 

.0018 

. 0181 

. 9819 

33 

12 

52 

28 

. 6047 

. 3953 

6.538 

. 6058 

6.507 

.0018 

. 0183 

. 9817 

32 

8 

56 

29 

. 6076 

. 3924 

6.459 

. 6087 

6.428 

.0018 

. 0185 

. %115 

31 

4 

14 

30 

.06105 

.93895 

16.380 

.06116 

16.350 

1.0019 

.00186 

.99813 

30 

46 

4 

31 

. 6134 

. 3866 

6.303 

. 6145 

6.272 

.0019 

. 0188 

. 9812 

29 

56 

8 

32 

. 6163 

. 3837 

6.226 

. 6175 

6.195 

.0019 

. 0190 

. 9810 

28 

52 

12 

33 

. 6192 

. 3808 

6.150 

. 6204 

6.119 

.0019 

. 0192 

. 9808 

27 

48 

16 

31 

. 6221 

. 3777 

6.075 

. 6233 

6.043 

.0019 

. 0194 

. 9806 

26 

44 

20 

35 

.06250 

.93750 

16.000 

.06262 

15.969 

1.0019 

.00195 

.99804 

25 

40 

24 

36 

. 6279 

. 3721 

5.926 

. 6291 

5.894 

.0020 

. 0197 

. 9803 

21 

36 

28 

37 

. 6308 

. 3692 

5.853 

. 6321 

5.821 

.0020 

. 0199 

. 9801 

23 

32 

32 

38 

. 6337 

. 3663 

5.780 

. 6350 

5.748 

.0020 

. 0201 

. 9799 

22 

28 

36 

39 

. 6366 

. 3634 

5.708 

. 6379 

5.676 

.0020 

. 0203 

. 9797 

21 

24 

40 

40 

.06395 

.93605 

15.637 

.06408 

15.605 

1.0020 

.00205 

.99795 

20 

20 

44 

41 

. 6424 

. 3576 

5.566 

. 6437 

5.534 

.0021 

. 0206 

. 9793 

19 

16 

48 

42 

. 6453 

. 3517 

5.496 

. 6467 

5.464 

.0021 

. 0208 

. 9791 

18 

12 

52 

43 

. 6482 

. 3518 

5.427 

. 6496 

5.394 

.0021 

. 0210 

. 9790 

17 

8 

56 

44 

. 6511 

. 3489 

5.358 

. 6525 

5.325 

.0021 

. 0212 

. 9788 

16 

4 

15 

45 ! 

.06540 

.93460 

15.290 

.06554 

15.257 

1.0021 

.00214 

.99786 

15 

45 

4 

46 

. 6569 

. 3431 

4.222 

. 6583 

5.189 

.0022 

. 0216 

. 9784 

14 

56 

8 

47 

. 6598 

. 3402 

5.155 

. 6613 

5.122 

.0022 

. 0218 

. 9782 

13 

52 

12 

48 

. 6627 

. 3373 

5.089 

. 6642 

5.056 

.0022 

. 0220 

. 9780 

12 

48 

16 

49 

. 6656 

. 3343 

5.023 

. 6671 

4.990 

.0022 

. 0222 

. 9778 

11 

44 

20 

50 

.06685 

.93314 

14.958 

.06700 

14.924 

1.0022 

.00224 

.99776 

10 

40 

24 

51 

. 6714 

. 3285 

4.893 

. 6730 

4.860 

.0023 

. 0226 

. 9774 

9 

36 

28 

52 

. 6743 

. 3256 

4.829 

. 6759 

4.795 

.0023 

. 0228 

. 9772 

8 

32 

31 

53 

. 6772 

. 3227 

4.765 

. 6788 

4.732 

.0023 

. 0230 

. 9770 

7 

28 

36 

54 

. 6801 

. 3198 

4.702 

. 6817 

4.668 

.0023 

. 0231 

. 9768 

6 

24 

40 

55 

.06830 

.93169 

14.640 

.06846 

14.606 

1.0023 

.00233 

.99766 

5 

20 

41 

56 

. 6859 

. 3140 

4.578 

. 6876 

4.544 

.0024 

. 0235 

. 9764 

4 

16 

48 

57 

. 6888 

. 3111 

4.517 

. 6905 

4.482 

.0024 

. 0237 

. 9762 

3 

12 

52 

58 

. 6918 

. 3082 

4.456 

. 6934 

4.421 

.0024 

. 0239 

. 9760 

2 

8 

56 

59 

. 6947 

. 3053 

4.395 

. 6963 

4.361 

.0024 

. 0241 

. 9758 

1 

4 

16 

60 

. 6976 

. 3024 

4.335 

. 6993 

4.301 

.0024 

. 0243 

. 9756 

0 

44 

M.S. 

M 

Cosine. 

Vrs.Sin- 

Seeante. 

Co tang.‘Tangent. 

Coseo’nt 

Vrs.Cos 

Sine. 

M M.S. 

! c h ! 

93° 




Natural. 



86°j 

5 h 












































Natural Lines. 


213 


O h 

4° 

Natural Trigonometrical Functions. 

175° 

ll h 

M. S 

M 

Sine. 

Vrs.Cos. 

ICoseo'nte 

Tang. 

Cotang. 

| Secante. 

iVrs. Sin 

Cosine. 

M 

M.S. 

16 

0 

.06976 

.93024 

14.335 

.06993 

11.301 

1.0024 

.00243 

.99756 

60 

44 

4 

1 

. 7005 

. 2995 

4.276 

. 7022 

4.241 

.0025 

. 0240 

. 9754 

59 

66 

8 

2 

. 7034 

. 2966 

4.217 

. 7051 

4.182 

.0025 

. 0243 

. 9752 

58 

52 

12 

3 

. 7063 

. 2937 

4.159 

. 7080 

4.123 

.0025 

. 0250 

. 9750 

57 

48 

16 

4 

. 7092 

. 2908 

4,101 

. 7110 

4.065 

.0025 

. 0252 

. 9748 

56 

44 

20 

5 

.07121 

.92879 

14.043 

.07139 

14.008 

1.0025 

.00254 

.99746 

55 

40 

24 

6 

. 7150 

. 2850 

3.986 

. 7168 

3.951 

.0026 

. 0256 

. 9744 

64 

36 

28 

7 

. 7179 

. 2821 

3.930 

. 7197 

3.894 

.0020 

. 0258 

. 9742 

53 

32 

32 

8 

. 7208 

. 2792 

3.874 

. 7226 

3.838 

.0026 

. 0260 

. 9740 

62 

23 

36 

9 

. 7237 

. 2763 

3.818 

. 7256 

3.782 

.0026 

. 0262 

. 9738 

51 

24 

40 

10 

.07-66 

92734 

13.763 

.07285 

13.727 

1.0026 

.00264 

.99736 

50 

20 

44 

11 

. 7295 

. 2705 

3.708 

. 7314 

3.672 

.0027 

. 0266 

. 9733 

49 

16 

48 

12 

. 7324 

. 2676 

3.654 

. 7343 

3.617 

.0027 

. 0268 

. 9731 

48 

12 

52 

13 

. 7353 

. 2647 

3.600 

. 7373 

3.563 

.0027 

. 0271 

. 9729 

47 

8 

56 

14 

. 7382 

. 2618 

3.547 

. 7402 

3.510 

.0027 

. 0273 

. 9727 

46 

4 

17 

15 

.07411 

.92589 

13.494 

.07431 

13.457 

1.0027 

.00275 

.99725 

45 

43 

4 

16 

. 7440 

. 2560 

3.441 

. 7460 

3.404 

.0028 

. 0277 

. 9723 

44 

56 

8 

17 

. 7469 

. 2531 

3.389 

. 7490 

3.351 

.0028 

. 0279 

. 9721 

43 

52 

12 

18 

. 7498 

. 2502 

3.337 

. 7519 

3.299 

.0028 

. 0281 

. 9718 

42 

48 

16 

19 

. 7527 

. 2473 

3.286 

. 7548 

3.248 

.0028 

. 0284 

. 9716 

41 

44 

20 

20 

.07556 

.92441 

13.235 

.07577 

13.197 

1.0029 

.00286 

.99714 

40 

40 

24 

21 

. 7585 

. 2415 

3.184 

. 7607 

3.146 

.0029 

. 0288 

. 9712 

39 

36 

28 

22 

. 7614 

. 2386 

3.134 

. 7636 

3.096 

.0029 

. 0290 

. 9710 

38 

32 

32 

23 

. 7643 

. 2357 

3.084 

. 7665 

3.046 

.0029 

. 0292 

. 9707 

37 

28 

36 

24 

. 7672 

. 2328 

3.034 

. 7694 

2.996 

.0029 

. 0295 

. 9705 

36 

24 

40 

25 

.07701 

.92299 

12.985 

.07724 

12.947 

1.0030 

.00297 

.99703 

35 

20 

44 

26 

. 7730 

. 2270 

2.937 

. 7753 

2.898 

.0030 

. 0299 

. 9701 

34 

16 

48 

27 

. 7759 

. 2241 

2.888 

. 7782 

2.849 

.0030 

. 0301 

. 9698 

33 

12 

52 

28 

. 7788 

. 2212 

2.840 

. 7812 

2.801 

.0030 

. 0304 

. 9696 

32 

8 

56 

29 

. 7817 

. 2183 

2.793 

. 7841 

2.754 

.0031 

. 0306 

. 9694 

31 

4 

IS 

30 

.07846 

.92154 

12.745 

.07870 

12.706 

1.0031 

.00308 

.99692 

30 

14 

4 

31 

. 7875 

. 2125 

2.698 

. 7899 

2.659 

.0031 

. 0310 

. 9689 

29 

56 

8 

32 

. 7904 

. 2096 

2.652 

. 7929 

2.612 

.0031 

. 0313 

. 9687 

28 

52 

12 

33 

. 7933 

. 2067 

2.606 

. 7958 

2.566 

.0032 

. 0315 

. 9685 

27 

48 

16 

34 

. 7962 

. 2038 

2.560 

. 7987 

2.520 

.0032 

. 0317 

. 9682 

26 

44 

20 

35 

.07991 

.92009 

12.514 

.0S016 

12.474 

1.0032 

.00320 

.99680 

25 

40 

24 

36 

. 8020 

. 1980 

2.469 

. 8046 

2.429 

.0032 

. 0322 

. 9678 

24 

36 

28 

37 

. 8049 

. 1951 

2.424 

. 8-175 

2.384 

.0032 

. 0324 

. 9675 

23 

32 

32 

38 

. 8078 

. 19-2 

2.379 

. 8104 

2.339 

.0033 

. 0327 

. 9673 

22 

28 

36 

39 

. 8107 

. 1893 

2.335 

. 8134 

2.295 

.0033 

. 0329 

. 9671 

21 

21 

40 

40 

.08136 

.91864 

12.291 

.08163 

12.250 

1.0033 

.00531 

.99668 

20 

20 

44 

41 

. 8165 

. 1835 

2.248 

. 8192 

2.207 

.0933 

. 0334 

. 9666 

19 

16 

48 

42 

. 8194 

. 1806 

2.204 

. 8221 

2.163 

.0034 

. 0336 

. 9664 

18 

12 

52 

43 

. 8223 

. 1777 

2.161 

. 8251 

2.120 

.0034 

. 0339 

. 9661 

17 

8 

56 

44 

. 8252 

. 1748 

2.118 

. 8280 

2.077 

.0034 

. 0341 

. 9659 

16 

4 

19 

45 

.08281 

.91719 

12.076 

.08309 

12.065 

1.0034 

.00343 

.99656 

15 

41 

4 

46 

. 8310 

. 1690 

2.034 

. 8339 

1.992 

.0035 

. 0346 

. 9654 

14 

56 

8 

47 

. 8339 

. 1661 

1.992 

. 8368 

1.950 

.0035 

. 0348 

. 9652 

13 

52 

12 

48 

. 87:68 

. 1632 

1.950 

. 8397 

1.909 

.0035 

. 0351 

. 9649 

12 

48 

16 

49 

. 8397 

. 1603 

1.909 

. 8426 

1.867 

.0035 

. 0353 

. 9647 

11 

44 

20 

50 

.08426 

.91574 

11 868 

.08456 

11.826 

1.0036 

.00356 

.99644 

10 

40 

24 

51 

. 8455 

. 1545 

1.82S 

. 8485 

1.785 

.0036 

. 0358 

. 9642 

9 

36 

28 

52 

. 8484 

. 1516 

1.787 

. 8514 

1.745 

.0036 

. 0360 

. 9639 

8 

32 

32 

53 

. 8513 

. 1487 

1.747 

. 8544 

1.704 

.0036 

. 0363 

. 9637 

7 

28 

36 

54 

. 8542 

. 1458 

1.707 

. 8573 

1.664 

.0037 

. 0365 

. 9634 

6 

21 

40 

55 

.08571 

.91429 

11.668 

.08602 

11.625 

1.0037 

.00368 

.99632 

5 

20 

44 

56 

. 8600 

. 1400 

1.628 

. 8632 

1.585 

.0037 

. 0370 

. 9629 

4 

16 

48 

57 

. 8629 

. 1371 

1.589 

. 8661 

1.546 

.0037 

0373 

. 9627 

3 

12 

52 

58 

. 8658 

. 1342 

1.550 

. 8690 

1.507 

.0038 

. 0375 

. 9624 

2 

8 

56 

59 

. 8687 

. 1313 

1.512 

. 8719 

1.468 

.0038 

. 0378 

. 9622 

1 

4 

30 

60 

. 8715 

. 1284 

1.474 

* 8749 

1.430 

.0038 

. 0380 

. 9619 

0 

40 

M.S.j M ! 

6 h ;94° 

Cosine. 

Vrs.Sin.! 

Secante- 

Cotang. 

Natti 

Tangent. 

tral. 

Cosec'nt < Vvs. Cosl 

Sine. 

GO 
Ci s 

°. 

M.S. 

5 h 































214 


Natural Lines. 


o h 

5° 

Natural Trig 

•on©metrical Functions. 

174° 

ll h 

M. S. 

M 

Sine. 

Vrs.Cos. 

iCosec'nte 

Tang. 

Cotaug. 

Secantc 

Vrs. Sin 

Cosine. 

M 

M.S. 

30 

0 

.08715 

.91284 

11.474 

.08749 

11.430 

1.0038 

.00380 

.99619 

60 

40 

4 

1 

. 8744 

. 1255 

1.436 

. 8778 

1.392 

.0038 

. 0383 

. 9617 

£9 

56 

8 

2 

. 8773 

. 1226 

1.398 

. 8807 

1.354 

.0039 

. 0386 

. 9614 

68 

52 

12 

3 

. 8802 

. 1197 

1.360 

. 8837 

1.316 

.0039 

. 0388 

. 9612 

57 

48 

16 

4 

. 8831 

. 1168 

1.S23 

. 8866 

1.279 

.0039 

. 0391 

. 9609 

66 

44 

20 

5 

.08860 

.91139 

11.286 

.08895 

11.242 

1.003J 

.00393 

.99607 

55 

40 

24 

6 

. 8889 

. 1110 

1.249 

. 8925 

1.205 

.0040 

. 0396 

. 9604 

54 

36 

28 

7 

. 8918 

. 1082 

1.213 

. 8954 

1.168 

.0040 

. 0398 

. 9601 

53 

32 

32 

8 

. 8947 

. 1053 

1.176 

. 8983 

1.132 

.0040 

. 0491 

. 9599 

52 

28 

36 

9 

. 8976 

. 1024 

1.140 

. 9013 

1.095 

.0040 

. 0404 

. 9596 

51 

24 

40 

10 

.09005 

.90995 

11.104 

.09042 

11.059 

1.0041 

.00406 

.99594 

50 

20 

44 

11 

. 9034 

. 0966 

1.069 

. 9071 

1.024 

.0041 

. 0409 

. 9591 

49 

16 

48 

12 

. 9063 

. 0937 

1033 

. 9101 

0.988 

.0041 

. 0411 

. 9588 

48 

12 

52 

13 

. 9092 

. 0908 

0.998 

. 9130 

0.953 

.0041 

. 0414 

. 9586 

47 

8 

56 

14 

. 9121 

. 0879 

0.963 

. 9159 

0.918 

.0042 

. 0417 

. 9583 

46 

4 

31 

15 

.09150 

.90850 

10.929 

.09189 

10.883 

1.0042 

.00419 

.99580 

45 

39 

4 

16 

. 9179 

. 0821 

0.894 

. 9218 

0.848 

.0042 

. 0422 

. 9578 

44 

56 

8 

17 

. 9208 

. 0792 

0.860 

. 9247 

0.814 

.0043 

. 0425 

. 9575 

43 

52 

12 

IS 

. 9237 

. 0763 

0.826 

. 9277 

0.780 

.0043 

. 0427 

. 9572 

42 

48 

16 

19 

. 9266 

. 0734 

0.792 

. 9306 

0.746 

.0043 

. 0430 

. 9570 

41 

44 

20 

20 

.09295 

.90705 

10.758 

.09335 

10.712 

1.0043 

.00433 

.99567 

40 

40 

24 

21 

. 9324 

. 0676 

0.725 

. 9365 

0.678 

.0044 

. 0436 

. 9564 

39 

36 

28 

22 

. 9353 

. 0647 

0.692 

. 9394 

0.645 

.0044 

. 0438 

. 9562 

38 

32 

32 

23 

. 9382 

. 0618 

0.659 

. 9423 

0.612 

.0044 

. 0441 

. 9559 

37 

28 

36 

24 

. 9411 

. 0589 

0 626 

. 9453 

0.579 

.0044 

. 0444 

. 9556 

36 

24 

40 

25 

.09440 

.90560 

10.593 

.09482 

10.546 

1.0045 

.00446 

.99553 

35 

20 

44 

26 

. 9469 

. 0531 

0.561 

. 9511 

0.514 

.0045 

. 0449 

. 9551 

34 

16 

48 

27 

. 9498 

. 0502 

0 529 

. 9541 

0.481 

.0045 

. 0452 

. 9548 

33 

12 

52 

28 

. 9527 

. 0473 

0.497 

. 9570 

0.449 

.0046 

. 0455 

. 9545 

32 

8 

56 

29 

. 9556 

. 0444 

0.465 

. 9599 

0.417 

.0016 

. 0458 

. 9542 

31 

4 

33 

30 

.09584 

.90415 

10.433 

.09629 

10.385 

1.0046 

.00460 

.99540 

30 

38 

4 

31 

. 9613 

. 0386 

0.402 

. 9658 

0.354 

.0046 

. 0463 

. 9537 

29 

£6 

8 

32 

. 9642 

. 0357 

0.371 

. 9688' 

0.322 

.0047 

. 0466 

. 9534 

28 

52 

12 

33 

. 9671 

. 0328 

0.340 

. 9717 

0.291 

.0047 

. 0469 

. 9531 

27 

48 

16 

34 

. 9700 

. 0300 

0.309 

. 9746 

0.260 

.0047 

. 0472 

. 9528 

26 

44 

20 

35 

.09729 

.90271 

10.278 

.09776 

10.229 

1.0048 

.00474 

.99525 

25 

40 

24 

36 

. 9758 

. 0242 

0.248 

. 9805 

0.199 

.004S 

. 0477 

. 9523 

24 

36 

28 

37 

. 9787 

. 0213 

0.217 

. 9834 

0.168 

.0048 

. 0480 

. 9520 

23 

32 

32 

38 

. 9816 

. 0184 

0.187 

. 9864 

0.138 

.0048 

. 0183 

. 9517 

22 

28 

£6 

39 

. 9845 

. 0155 

0.157 

. 9893 

0.108 

.0049 

. 0486 

. 9514 

21 ' 

21 

40 

40 

.09874 

.90126 

10.127 

.09922 

10.978 

1.0049 

.00489 

.99511 

20 

20 

44 

41 

. 9903 

. 0097 

0.098 

. 9952 

0.048 

.0649 

. 0491 

. 9508 

19 

16 

48 

42 

. 9932 

. 0068 

0.068 

. 9981 

0.019 

.0050 

. 0494 

. 9505 

18 

12 

52 

43 

. 9961 

. 0039 

0.039 

.10011 

9.9893 

.0050 

. 0497 

. 9503 

17 

8 

56 

44 

. 9990 

. 0010 

0.010 

.10040 

.9601 

.0050 

. 0500 

. 9500 

16 

4 

33 

45 

.10019 

.89981 

9.9812 

.10069 

.9310 

1.0050 

.00503 

.99497 

15 

37 

4 

46 

. 0048 

. 9952 

.9525 

. 0099 

.9021 

.0051 

. 0506 

. 9494 

14 

56 

8 

47 

. 0077 

. 9923 

.9239 

. 0128 

.8734 

.0051 

. 0509 

. 9491 

13 

52 

12 

48 

. 0106 

. 9894 

.8955 

. 0158 

.8448 

.0051 

. 0512 

. 9488 

12 

48 

16 

49 

. 0134 

. 9865 

.8672 

. 0187 

.8164 

.0052 

. 0515 

. 9485 

11 

44 

20 

50 

.10163 

.89836 

9.8591 

.10216 

9.7 882 

1.0052 

.00518 

.99482 

10 

40 

24 

51 

. 0192 

. 9807 

.8112 

. 0246 

.7601 

.0052 

. 0521 

. 9479 

9 

36 

28 

52 

. 0221 

. 9779 

.7834 

. 0275 

.7322 

.0053 

. 0524 

. 9476 

8 

32 

32 

53 

. 0250 

. 9750 

.7558 

. 0305 

.7044 

.0053 

. 0527 

. 9473 

7 

28 

36 

54 

. 0279 

. 9721 

.7283 

. 0334 

.6768 

.0053 

. 0530 

. 9470 

6 

24 

40 

55 

.10308 

.89692 

9.7010 

.10363 

9.6493 

1.0053 

.00533 

.99467 

5 

20 

4i 

56 

. 0337 

. 9663 

.6739 

. 0393 

.6220 

.9054 

. 0536 

. 9464 

4 

10 

48 

57 

. 0366 

. 9634 

.6469 

. 0422 

.5949 

.0054 

. 0539 

. 9461 

3 

12 

52 

58 

. 0395 

. 9605 

.6200 

. 0452 

.5679 

.0054 

. 0542 

. 9458 

2 

8 

56 

59 

. 0424 

. 9576 

.5933 

. 0481 

.5411 

.0055 

. 0545 

. 9455 

1 

4 

24 

60 

. 0453 

. 9547 

.5668 

. 0510 

.5144 

.0055 

. 0548 

. 9452 

0 

30 

M. S. 

M 

Cosiue. 

Vis. Sin. 

Secante. 

Cotang. .Tangent,. 

Cosee'nt t Vrs. Cos 

Sine. 

M 

M.S. 

6 h 

95° 




Natural. 



o 

.00 

5 h 










































Natural Lines, 


215 


O h 

6° 

Natural Trig 

onometrical 

Functions. 

o 

CO 

r- 

1—i 

ll h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante.'Vrs. Sin 

Cosine. 

M 

M.S. 

34 

0 

.10453 

.89547 

9.5668 

.10510 

9.5144 

1.0055 

.00548 

.99452 

60 

3G 

4 

1 

. 0482 

. 9518 

.5404 

. 0540 

.4878 

.((055 

. 0551 

. 9449 

59 

56 

8 

2 

. 0511 

. 9489 

.5141 

. 0569 

.4614 

.0056 

. 0554 

. 9446 

58 

52 

12 

3 

. 0540 

. 9460 

.4880 

. 0599 

.4351 

.0056 

. 0557 

. 9443 

57 

48 

16 

4 

. 0568 

. 9431 

.4620 

. 0628 

.4090 

.0076 

. 0560 

. 9440 

56 

44 

20 

5 

.10597 

.89402 

9.4362 

.10657 

9.3831 

1.0057 

.00563 

.99437 

55 

40 

24 

6 

. 0626 

. 9373 

.4105 

. 0687 

.3572 

.0057 

. 0566 

. 9434 

54 

36 

28 

7 

. 0055 

. 9345 

.3850 

. 0716 

.3315 

.0057 

. 0569 

. 9431 

53 

32 

32 

8 

. 0084 

. 9316 

.3596 

. 0746 

.3060 

.0057 

. 0572 

. 9428 

52 

28 

36 

9 

. 0713 

. 9287 

.3343 

. 0775 

.2806 

.0058 

. 0575 

. 9424 

51 

24 

40 

10 

.10742 

.89258 

9.3092 

.10805 

9.2553 

1.0058 

.00579 

.99421 

50 

20 

41 

11 

. 0771 

. 9229 

.2842 

. 0834 

.2302 

.0058 

. 0582 

. 9418 

49 

16 

48 

12 

. 0800 

. 9200 

.2593 

. 0863 

.2051 

.0059 

. 0585 

. 9415 

48 

12 

52 

13 

. 0829 

. 9171 

.2346 

. 0893 

.1803 

.0059 

. 0588 

. 9412 

47 

8 

56 

14 

. 0858 

. 9142 

.2100 

. 0922 

.1555 

.0059 

. 0591 

. 9409 

46 

4 

35 

15 

.10887 

.89113 

9.1855 

.10952 

9.13(19 

1.0060 

.00594 

.99406 

45 

35 

4 

16 

. 0916 

. 9081 

.1612 

. 0981 

.1064 

.0060 

. 0597 

. 9402 

44 

56 

8 

17 

. 0944 

. 9055 

.1370 

. 1011 

.0821 

.0060 

. 0601 

. 9399 

43 

52 

12 

18 

. 0973 

. 9026 

.1129 

. 1040 

.0579 

.0061 

. 0604 

. 9396 

42 

48 

16 

19 

. 1002 

. 8998 

.0890 

. 1069 

.0338 

.0061 

. 0607 

. 9393 

41 

44 

20 

20 

.11(131 

.S8969 

9.0651 

.11099 

9.0098 

1.0061 

.00! 10 

.99390 

40 

40 

24 

21 

. 1060 

. 8940 

.0414 

. 1128 

8.9860 

.0062 

. 0613 

. 9386 

39 

36 

28 

22 

. 1089 

. 8911 

.0179 

. 115S 

.9623 

.0062 

. 0617 

. 9383 

38 

32 

32 

23 

. 1118 

. 8882 

.9944 

. 1187 

.9387 

.0062 

. 0620 

. 9380 

37 

28 

36 

24 

. 1147 

. 8853 

8.9711 

. 1217 

.9152 

.0063 

. 0623 

. 9377 

36 

24 

40 

25 

.11176 

.88824 

8.9479 

.11246 

8.8918 

1.0063 

.00626 

.99373 

35 

20 

44 

26 

. 1205 

. 8795 

.9248 

. 1276 

.8686 

.0063 

. 0630 

. 9370 

34 

16 

48 

27 

. 1234 

. 8766 

.9018 

. 1305 

.8455 

.0064 

. 0633 

. 9367 

33 

12 

52 

28 

. 1202 

. 8737 

.8790 

. 1335 

.8225 

.0064 

. 0636 

. 9364 

32 

8 

56 

29 

. 1291 

. 8708 

.8563 

. 1364 

.7996 

.0064 

. 0639 

. 9360 

31 

4 

26 

30 

.11320 

.88680 

8.8337 

.11393 

8.7769 

1.0065 

.00643 

.99357 

30 

34 

4 

31 

. 1349 

. 8651 

.8112 

. 1423 

.7542 

.0065 

. 0646 

. 9354 

29 

56 

8 

32 

. 1378 

. 8622 

.7888 

. 1452 

.7317 

.0065 

. 0649 

. 9350 

28 

52 

12 

33 

. 1407 

. 8593 

.7665 

. 1482 

.7093 

.0066 

. 0653 

. 9347 

27 

48 

16 

34 

. 1436 

. 8564 

.7444 

. 1511 

.6870 

.0066 

. 0656 

. 9344 

26 

44 

20 

35 

.11405 

.88535 

8.7223 

.11541 

8.6648 

1.0066 

.00659 

.99341 

25 

40 

24 

36 

. 1494 

. 8506 

.7004 

. 1570 

.6427 

.0067 

. 0663 

. 9337 

24 

36 

28 

37 

. 1523 

. 8477 

.6786 

. 1600 

.6208 

.0067 

. 0666 

. 9334 

23 

32 

32 

38 

. 1551 

. 8448 

.6569 

. 1629 

.5989 

.0067 

. 0669 

. 9330 

22 

28 

36 

39 

. 1580 

. 8420 

.6353 

. 1659 

.5772 

.0068 

. 0673 

. 9327 

21 

24 

40 

40 

.11609 

.88391 

8.6138 

.11688 

8.5555 

1.0068 

.00676 

.99324 

20 

20 

44 

41 

. 1638 

. 8362 

.5924 

. 1718 

.5340 

.0068 

. 0679 

. 9320 

19 

16 

48 

42 

. 1667 

. 8333 

.5711 

. 1747 

.5126 

.0069 

. 0683 

. 9317 

18 

12 

52 

43 

. 1696 

. 8304 

.5499 

. 1777 

.4913 

.006) 

. 0686 

. 9314 

17 

8 

56 

44 

. 1725 

. 8272 

.5289 

. 1806 

.4701 

.0069 

. 0690 

. 9310 

16 

4 

27 

45 

.11754 

.88246 

8.5079 

.11836 

8.4489 

1.0070 

.00693 

99307 

15 

33 

4 

46 

. 1783 

. 8217 

.4871 

. 1865 

.4279 

.0070 

. 0696 

. 9303 

14 

56 

8 

47 

. 1811 

. 8188 

.4663 

. 1895 

.4070 

.0070 

. 0700 

. 9300 

13 

52 

12 

48 

. 1840 

. 8160 

.4457 

. 1924 

.3S62 

.0071 

. 0703 

. 9296 

12 

48 

16 

49 

. 1869 

. 8131 

.4251 

. 1954 

.3655 

.0071 

. 0707 

. 9293 

11 

44 

20 

50 

.11898 

.88102 

8.4046 

.11983 

8.3449 

1.0071 

.00710 

.99290 

10 

40 

24 

51 

. 1927 

. 8073 

.3843 

. 2013 

.3244 

.0072 

. 0714 

. 9286 

9 

36 

28 

52 

. 1956 

. 8044 

.3640 

. 2042 

■ .3040 

.0072 

. 0717 

. 9283 

8 

32 

32 

53 

. 1985 

. 8015 

.3439 

. 2072 

.2837 

.0073 

. 0721 

. 9279 

7 

28 

36 

54 

. 2014 

. 79S6 

.3238 

. 2101 

.2^35 

.0073 

. 0724 

. 9276 

6 

24 

40 

55 

.12042 

.87957 

8.3039 

.12131 

8.2434 

1.0073 

.00728 

.99272 

5 

20 

44 

56 

. 2071 

. 7928 

.2840 

. 2160 

.2234 

.0074 

. 0731 

. 9269 

4 

16 

48 

57 

. 2100 

. 7900 

.2642 

. 2190 

.2035 

.0074 

. 0735 

. 9265 

3 

12 

52 

58 

. 2129 

. 7871 

.2446 

. 2219 

.1837 

.0074 

. 0738 

. 9262 

2 

8 

56 

59 

. 2158 

. 7842 

.2250 

. 2249 

.1640 

.0075 

. 0742 

. 9258 

1 

4 

28 

60 

. 2187 

. 7813 

.2055 

. 2278 

.1443 

.0075 

. 0745 

. 9255 

0 

32 

M. S. 

M 

Cosine. 

Vra.Siu. 

Secante. 

Co tang. iTaugeut. 

Cosec'nt 

Vrs. Cos 

Sine. 

M 

M.S. 

6 h 

96° 




Natural. 




83°| 

5 h 





















216 


Natural Lines. 


o h 

7° 

Natural Trigonometrical Functions 

172° 

ll h 

w.s. 

M 

Sine. 

Vrs. Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secaute. 

Vrs. Sin j Cosine. 

M 

M.S. 

as 

0 

.12187 

.87813 

8.2055 

.12278 

8.1443 

1.0075 

.09745 

1 .99255 

60 

3 a 

4 

1 

. 2216 

. 7787 

.1861 

. 2308 

.1248 

.0075 

. 0749 

. 9251 

59 

5(5 

8 

2 

. 2245 

. 7755 

.1663 

. 2337 

.1053 

.0076 

. 0752 

. 9247 

58 

52 

12 

3 

. 2273 

. 7726 

.1476 

. 2367 

.0860 

.0076 

. 0756 

. 9244 

57 

48 

16 

4 

. 2302 

. 7697 

.1285 

. 2396 

.0667 

.0076 

. 0760 

. 9240 

56 

44 

20 

5 

.12331 

.87669 

8.1094 

.12426 

8.0476 

1.0077 

.00763 

.99237 

55 

40 

24 

6 

. 2360 

. 7640 

.0905 

. 2456 

• .0285 

.0077 

. 0767 

. 9233 

54 

36 

28 

7 

. 2389 

. 7611 

.0717 

. 2485 

.0095 

.0078 

. 0770 

. 9229 

53 

32 

32 

8 

. 2418 

. 7582 

.0529 

. 2515 

7.9906 

.0078 

. 0774 

. 9226 

52 

28 

36 

9 

. 2447 

. 7553 

.0342 

. 2544 

7.9717 

.0078 

. 0778 

. 9222 

51 

24 

40 

10 

.12476 

.87524 

8.0156 

.12574 

7.9530 

1.0079 

.00781 

.99219 

50 

20 

44 

11 

. 2504 

. 7495 

7.9971 

. 2603 

.9344 

.0079 

. 0785 

. 9215 

49 

16 

48 

12 

. 2533 

. 7467 

.9787 

. 2633 

.9158 

.0079 

. 0788 

. 9211 

48 

12 

52 

13 

. 2562 

. 7438 

.9604 

. 2662 

.8 <73 

.0080 

. 0792 

. 9208 

47 

8 

56 

14 

. 2591 

. 7409 

.9421 

. 2692 

.8789 

.0080 

. 0796 

..9201 

46 

4 

39 

15 

.12620 

.87380 

7.9240 

.12722 

7.8606 

1.0080 

.00799 

.99200 

45 

31 

4 

16 

. 2649 

. 7351 

.9059 

. 2751 

.8424 

.0081 

. 0803 

. 9197 

44 

56 

8 

17 

. 2678 

. 7322 

.8879 

. 2781 

.8243 

.0081 

. 0807 

. 9193 

43 

52 

12 

18 

. 2706 

. 7293 

.8700 

. 2810 

.8062 

.0082 

. 0810 

. 9189 

42 

48 

16 

19 

. 2735 

. 7265 

.8522 

. 2840 

.7882 

.0082 

. 0814 

. 9186 

41 

44 

20 

20 

.12764 

.87236 

7.8344 

.12869 

7.7703 

1.0082 

.00818 

.99182 

40 

41 

24 

21 

. 2793 

. 7207 

.8168 

. 2899 

.7525 

.0083 

. 0822 

. 9178 

39 

36 

28 

22 

. 2822 

. 7178 

.7992 

. 2928 

.7348 

.0083 

. 0825 

. 9174 

38 

32 

32 

23 

. 2851 

. 7149 

.7817 

.2958 

.7171 

.0084 

. 0829 

. 9171 

37 

28 

36 

24 

. 2879 

. 7120 

.7642 

. 2988 

.6996 

.0084 

. 0833 

. 9167 

36 

24 

40 

25 

.12908 

.87091 

7.7469 

.13017 

7.6821 

1.0084 

.00837 

.99163 

35 

20 

44 

26 

. 2937 

. 7063 

.7296 

. 3047 

.6646 

.0085 

. 0840 

. 9160 

34 

16 

48 

27 

. 2966 

. 7034 

.7124 

. 3076 

.6473 

.0085 

. 0844 

. 9156 

33 

12 

52 

28 

. 2995 

. 7005 

.6953 

. 3106 

.6300 

.0085 

. 0S48 

. 9152 

32 

8 

56 

29 

. 3024 

. 6976 

.6783 

. 3136 

.6129 

.0086 

. 0852 

. 9148 

31 

4 

30 

30 

.13053 

.86947 

7.6613 

.13165 

7.5957 

1.0086 

.Ot'&SS 

.99144 

30 

30 

4 

31 

. 3081 

. 6918 

.6444 

. 3195 

.5787 

.0087 

. 0859 

. 9141 

29 

56 

8 

32 

. 3110 

. 6890 

.6276 

. 3224 

.5617 

.0087 

. 0863 

. 9137 

28 

52 

12 

33 

. 3139 

. 6861 

.6108 

. 3254 

.5449 

.0087 

. 0867 

. 9133 

27 

48 

16 

34 

3168 

. 6832 

.5942 

. 3284 

.5280 

.0088 

. 0871 

. 9129 

26 

44 

20 

35 

.13197 

.86803 

7.5T76 

.13313 

7.5113 

1.00S8 

.00875 

.99125 

25 

40 

24 

36 

. 3226 

. 6774 

.5611 

. 3343 

.4946 

.0089 

. 0878 

. 9121 

24 

36 

28 

37 

. 3254 

. 6745 

.5446 

. 3372 

.4780 

.0089 

. 0882 

. 9118 

23 

32 

32 

38 

. 3283 

. 6717 

.52S2 

. 3402 

.4015 

.0089 

. 0886 

. 9114 

22 

28 

36 

39 

. 3312 

. 6688 

.5119 

. 3432 

.4451 

.0090 

. 0890 

. 9110 

21 

24 

40 

40 

.13341 

.86659 

7.4957 

.13461 

7.4287 

1.0090 

.00894 

.99106 

20 

20 

44 

41 

. 3370 

. 6630 

.4795 

. 3491 

.4124 

.0090 

. 0898 

. 9102 

19 

16 

48 

42 

. 3399 

. 6601 

.4634 

. 3520 

.3961 

.0091 

. 0902 

. 9098 

18 

12 

52 

43 

. 3427 

. 6572 

.4474 

. 3550 

.3800 

.0091 

. 0905 

. 9094 

1/ 

8 

56 

44 

. 3456 

. 6544 

.4315 

. 3580 

.3639 

.0092 

. 0909 

. 9090 

16 

4 

31 

45 

.13485 

.86515 

7.4156 

.13609 

7.3479 

1.0092 

.00913 

.99086 

15 

39 

4 

46 

. 3514 

. 6486 

.3998 

. 3639 

.3319 

.0092 

. 0917 

. 9083 

14 

56 

8 

47 

. 3543 

. 6457 

.3840 

. 3669 

.3160 

.0093 

. 0921 

. 9079 

13 

52 

12 

48 

. 3571 

. 6428 

.3683 

. 3698 

.3002 

.0093 

. 0925 

. 9075 

12 

98 

16 

49 

. 3600 

. 6400 

.3527 

. 3728 

.2844 

.0091 

. 0929 

. 9070 

11 

44 

20 

50 

.13629 

.86371 

7.3372 

.13757 

7.2687 

1.0094 

.00933 

.99067 

10 

40 

24 

51 

. 3658 

. 6342 

.3217 

. 3787 

.2531 

.0094 

. 0937 

. 9063 

9 

36 

28 

52 

. 3687 

. 6313 

.3063 

. 3817 

.2375 

.0095 

. 0941 

. 9059 

8 

32 

32 

53 

. 3716 

. 6284 

.2909 

. 3846 

.2220 

.0095 

. 0945 

. 9055 

7 

28 

36 

54 

. 3744 

. 6255 

.2757 

. 3S76 

.2066 

-.0096 

. 0949 

. 9051 

6 

24 

40 

55 

.13773 

.86227 

7.2604 

.13906 

7.1912 

1.0096 

.00953 

.99047 

5 

20 

44 

56 

. 3S02 

. 6198 

.2453 

. 3935 

.1759 

.0097 

. 0957 

. 9043 

4 

16 

48 

57 

. 3831 

. 6169 

.2302 

. 3965 

.1607 

.0097 

. 0961 

. 9039 

3 

12 

52 

58 

. 3860 

. 6140 

.2152 

. 3995 

.1455 

.0097 

. 0965 

. 9035 

2 

8 

56 

59 

. 3^88 

. 6111 

.2002 

. 4024 

.1304 

.0098 

. 0969 

. 9031 

1 

4 

34 

60 

. 3917 

. 6083 

.1853 

. 4054! 

.1154 

.0398 

. 0973 

. 9027 

0 

as 

M.S. 

6 U 

° 

Cosine. 

Vrs. Sin.! Secaute. 

Cotang.iTaugent. | 

Natural. 

Cosoo’iit 

Vr..Cos 

Sine. 1 

M 

82° 

M.S. 

5 11 































Natural Lines. 


217 


o h 

O 

00 

Natural Trig 

onometrical Functions. 

171° 

ll h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante.jVrs.Sin 

Cosine. 

M 

M.S. 

3.4 

0 

.13917 

.86083 

7.1853 

.14054 

7.1154 

1.009S 

.00973 

.99027 

60 

as 

4 

1 

. 3946 

. 6054 

.1704 

. 4084 

.1001 

.0099 

. 0977 

. 9023 

59 

56 

8 

2 

. 3975 

. 6025 

.1557 

. 4113 

.0854 

.0099 

. 0981 

. 9019 

58 

52 

12 

3 

. 4004 

. 5996 

.1409 

. 4143 

.0706 

.0099 

. 09 .'.5 

. 9015 

57 

48 

16 

4 

. 4032 

. 5967 

.1263 

. 4173 

.0558 

.0100 

. 09S9 

. 9010 

56 

44 

20 

5 

.14061 

.85939 

7.1117 

.14202 

7.0410 

1.0100 

.00993 

.99006 

£5 

40 

24 

6 

. 4090 

. 5910 

.0972 

. 4232 

.0261 

.0101 

. 0998 

. 9002 

54 

36 

28 

7 

. 4119 

. 5881 

.0827 

. 4262 

.0117 

.0101 

. 1002 

. 8998 

53 

32 

32 

8 

. 4148 

. 5852 

.0683 

. 4291 

6.9372 

.0102 

. 1006 

. 8994 

52 

28 

36 

9 

. 4176 

. 5823 

.0539 

. 4321 

6.9827 

.0102 

. 1010 

. 8990 

51 

24 

40 

10 

.14205 

.85795 

7.0396 

.14351 

6.9682 

1.0102 

.01014 

.98986 

50 

20 

41 

11 

. 4234 

. 5766 

.0254 

. 43S0 

.9538 

.0103 

. 1018 

. 8982 

49 

16 

48 

12 

. 4263 

. 5737 

.0112 

. 4110 

.9395 

.0103 

. 1022 

. 8978 

48 

12 

52 

13 

. 4292 

. 5708 

6.9971 

. 4440 

.9252 

.0104 

. 1026 

. 8973 

47 

8 

56 

14 

. 4320 

. 5679 

6.9830 

. 4470 

.9110 

.0101 

. 1031 

. 8969 

46 

4 

33 

15 

.14349 

.85651 

6.9690 • 

.14499 

6.8969 

1.0104 

.01035 

.98965 

45 

37 

4 

16 

. 4378 

. 5622 

.9550 

. 4529 

.8828 

.0105 

. 1039 

. 8961 

44 

56 

8 

17 

. 4407 

. 5593 

.9411 

. 4559 

.8687 

.0105 

. 1043 

. 8957 

43 

52 

12 

IS 

. 4436 

. 5564 

.9273 

. 4588 

.8547 

.0106 

. 1047 

. 8952 

42 

48 

16 

19 

. 4464 

. 5536 

.9135 

. 4618 

.8408 

.0106 

. 1052 

. 8948 

41 

44 

20 

20 

.14493 

.85507 

6.8998 

.14648 

6.8269 

1.0107 

.01056 

.98944 

40 

40 

24 

21 

. 4522 

. 5478 

.8861 

. 4677 

.8131 

.0107 

. 1060 

. 8940 

39 

36 

28 

22 

. 4551 

. 5449 

.8725 

. 4707 

.7993 

.0107 

. 1064 

. 8936 

38 

32 

32 

23 

. 4579 

. 5420 

.8589 

. 4737 

.7856 

.0108 

. 1068 

. 8931 

37 

28 

36 

24 

. 4608 

. 5392 

.8454 

. 4767 

.7720 

.0108 

. 1073 

. 8927 

36 

24 

40 

25 

.14637 

.85363 

6.8320 

.14796 

6.7584 

1.0109 

.01077 

.98923 

35 

20 

44 

26 

. 4666 

. 5334 

.8185 

. 4826 

.7448 

.0109 

. 1081 

. 8919 

34 

16 

48 

27 

. 4695 

. 5305 

.8052 

. 4856 

.7313 

.0110 

. 1085 

. 8914 

33 

12 

52 

28 

. 4723 

. 5277 

.7919 

. 4886 

.7179 

.0110 

. 1090 

. 8910 

32 

8 

56 

29 

. 4752 

. 5248 

.7787 

. 4915 

.7045 

.0111 

. 1094 

. 8906 

31 

4 

34 

30 

.14781 

.85219 

6.7655 

.14945 

6.6911 

1.0111 

.01098 

.98901 

30 

36 

4 

31 

. 4810 

. 5190 

.7523 

. 4975 

.6779 

.0111 

. 1103 

. 8897 

29 

56 

8 

32 

. 4838 

. 5161 

.7392 

. 5004 

.6646 

.0112 

. 1107 

. 8893 

28 

52 

12 

33 

. 4867 

. 5133 

.7262 

. 5034 

.6514 

.0112 

. 1111 

. 8889 

27 

48 

16 

34 

. 4896 

. 5104 

.7132 

. 5064 

.6383 

.0113 

. 1116 

. 8884 

26 

44 

20 

35 

.14925 

.85075 

6.7003 

.15094 

6.6252 

1.0113 

.01120 

.98880 

25 

40 

24 

36 

. 4953 

. 5046 

.6874 

. 5123 

.6122 

.0114 

. 1124 

. 8876 

24 

36 

28 

37 

. 4982 

. 5018 

.6745 

. 5153 

.5992 

.0114 

. 1129 

. 8871 

23 

32 

32 

38 

. 5011 

. 4989 

.6617 

. 5183 

.5863 

.0115 

. 1133 

. 8867 

22 

28 

36 

39 

. 5040 

. 4960 

.6490 

. 5213 

.5734 

.0115 

. 1137 

. 8862 

21 

24 

40 

40 

.15068 

.84931 

6.6363 

.15243 

6.5605 

1.0115 

.01142 

.98858 

20 

20 

44 

41 

. 5097 

. 4903 

.6237 

. 5272 

.5478 

.0116 

. 1146 

. 8854 

19 

16 

48 

42 

. 5126 

. 4874 

.6111 

. 5302 

.5350 

.0116 

. 1151 

. 8849 

18 

12 

52 

43 

. 5155 

. 4845 

.5985 

. 5332 

.5223 

.0117 

. 1155 

. 8845 

17 

8 

56 

44 

. 5183 

. 4816 

.5860 

. 5362 

.5097 

.0117 

. 1159 

. 8840 

16 

4 

35 

45 

.15212 

.84788 

6.5736 

.15391 

6.4971 

1.0118 

.01164 

.98836 

15 

35 

4 

46 

. 5241 

. 4759 

.5612 

. 5421 

.4815 

.0118 

. 1168 

. 8832 

14 

56 

8 

47 

. 5270 

. 4730 

.5488 

. 5451 

.4720 

.0119 

. 1173 

. 8827 

13 

52 

12 

48 

. 5298 

. 4701 

.5365 

. 5481 

.4596 

.0119 

. 1177 

. 8823 

12 

48 

16 

49 

. 5328 

. 4672 

.5243 

. 5511 

.4472 

.0119 

. 1182 

. 8818 

11 

44 

20 

50 

.15356 

.84614 

6.5121 

.15540 

6.4348 

1.0120 

.01186 

.98814 

10 

40 

24 

51 

. 5385 

. 4615 

.4999 

. 5570 

.4225 

.0120 

. 1190 

. 8809 

9 

36 

28 

52 

. 5413 

. 4586 

.4878 

. 5600 

.4103 

.0121 

. 1195 

. 8805 

8 

32 

32 

53 

. 5442 

. 4658 

.4757 

. 5630 

.3980 

.0121 

. 1199 

. 8800 

7 

28 

36 

54 

. 5471 

. 4529 

.4637 

. 5059 

.3859 

.0122 

. 1204 

. 8796 

6 

24 

40 

55 

.15500 

.84500 

6.4517 

.15689 

6.3737 

1.0122 

.01208 

.98791 

5 

20 

44 

56 

. 5528 

. 4471 

.4398 

. 5719 

.3616 

.0123 

. 1213 

. 8787 

4 

16 

48 

57 

. 5557 

. 4413 

.4279 

. 5749 

.3496 

.0123 

. 1217 

. 8782 

3 

12 

52 

58 

. 5586 

. 4414 

.4160 

. 5779 

.3376 

.0124 

. 1222 

. 8778 

2 

8 

66 

59 

. 5615 

. 43S5 

.4042 

. 5809 

.3257 

.0124 

. 1227 

. 8773 

1 

4 

30 

60 

. 5643 

. 4356 

.3924 

. 5838 

.3137 

.0125 

. 1231 

. 8769 

0 

34 

M. S. 

M 

Cosine. 

Yrs.Sin. 

Secauto. 

Co tang. (Tangent. 

Cosec ut 1 

Sine, i 

Vrs.Cos! 

M 

il.S. 

6* 

98° 




Natural. 



Si 0 ! 

o h 

































218 


Natural Lines, 


o h 

9° 

Natural Trigonometrical Functions 

170° 

ll h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosee’nte 

Tang. 

Cotang. 

Secaute 

iVrs.Sin 

Cosine. 

M 

M.S. 

30 

0 

.15643 

.84356 

6.3924 

.15838 

6.3137 

1.0125 

.01231 

.98769 

60 

24 

4 

1 

. 5672 

. 4328 

.3807 

. 5868 

.3019 

.0125 

. 1236 

. 8704 

59 

66 

8 

2 

. 5701 

. 4299 

.3690 

. 5898 

.2901 

.0125 

. 1240 

. 8760 

58 

52 

12 

3 

. 5730 

. 4270 

.3574 

. 5928 

.2783 

.0126 

. 1245 

. 8755 

57 

48 

16 

4 

. 5758 

. 4242 

.3458 

. 5958 

.2665 

.0126 

. 1249 

. 8750 

56 

44 

20 

5 

.15787 

.84213 

6.3343 

.15987 

6.2548 

1.0127 

.01254 

.98746 

55 

40 

24 

6 

. 5816 

. 4184 

.3228 

. 6017 

.2432 

.0127 

. 1259 

. 8741 

54 

36 

28 

7 

. 5844 

. 4155 

.3113 

. 6017 

.2316 

.0128 

. 1263 

. 8737 

53 

32 

32- 

8 

. 5873 

. 4127 

.2999 

. 6077 

.2200 

.0128 

. 1268 

. 8732 

52 

28 

36 

9 

. 6902 

. 4098 

.2885 

. 6107 

.2085 

.0129 

. 1272 

. 8727 

51 

24 

40 

10 

.15931 

.84069 

6.2772 

.16137 

6.1970 

1.0129 

.01277 

.98723 

50 

20 

44 

11 

. 5959 

. 4041 

.2659 

. 6167 

.1856 

.0130 

. 1282 

. 8718 

49 

16 

48 

12 

. 5988 

. 4012 

.2546 

. 6196 

.1742 

.0130 

. 1286 

. 8714 

48 

12 

52 

13 

. 6017 

. 3983 

.2434 

. 6226 

.1628 

.0131 

. 1291 

. 8709 

47 

8 

56 

14 

. 6045 

. 3954 

.2322 

. 6256 

.1515 

.0131 

. 1296 

. 8704 

46 

4 

37 

15 

.16074 

.83926 

6.2211 

.16286 

6.1402 

1.0132 

.01300 

.98700 

45 

23 

4 

16 

. 6103 

. 3897 

.2100 

. 6316 

.1290 

.0132 

. 1305 

. 8695 

44 

56 

8 

17 

. 6132 

. 3868 

.1990 

. 6346 

.1178 

.0133 

. 1310 

. 8690 

43 

52 

12 

18 

. 6160 

. 3840 

.1880 

. 6376 

.1066 

.0133 

. 1314 

. 8685 

42 

48 

16 

19 

. 6189 

. 3811 

.1770 

. 6405 

.0955 

.0134 

. 1319 

. 8681 

41 

44 

20 

20 

.16218 

.83782 

6.1661 

.16435 

6.0844 

1.0134 

.01324 

.98670 

40 

40 

24 

21 

. 6246 

. 3753 

.1552 

. 6465 

.0734 

.0135 

. 1328 

. 8671 

39 

36 

28 

22 

. 6275 

. 3725 

.1443 

. 6495 

.0624 

.0135 

. 1333 

. 8667 

38 

32 

32 

23 

. 6304 

. 3696 

.1335 

. 6525 

.0514 

.0136 

. 1338 

. 8662 

37 

28 

36 

24 

. 6333 

. 3667 

.1227 

. 6555 

.0405 

.0136 

. 1343 

. 8057 

36 

24 

40 

25 

.16361 

.83639 

6.1120 

.16585 

6.0296 

1.0136 

.01317 

.98652 

35 

20 

44 

26 

. 6390 

. 3610 

.1013 

. 6615 

.0188 

.0137 

. 1352 

. 8648 

34 

16 

48 

27 

. 6419 

. 3581 

.0906 

. 6644 

.0080 

.0137 

. 1357 

. 8643 

33 

12 

52 

28 

. 6447 

. 3553 

.0800 

. 6674 

5.9972 

.0138 

. 1362 

. 8638 

32 

8 

56 

29 

. 6476 

. 3524 

.0694 

. 6704 

5.9865 

.0138 

. 1367 

. 8633 

31 

4 

38 

30 

.16505 

.83495 

6.0588 

.16734 

5.9758 

1.0139 

.01371 

.98628 

30 

22 

4 

31 

. 6533 

. 3466 

.0483 

. 6764 

.9651 

.0139 

. 1376 

. 8624 

29 

56 

8 

32 

. 6562 

. 3438 

.0379 

. 6794 

.9545 

.0140 

. 1381 

. 8619 

28 

52 

12 

33 

. 6591 

. 3109 

.0274 

. 6824 

.9439 

.0140 

. 1386 

. 8614 

27 

48 

16 

34 

. 6619 

. 3380 

.0170 

. 6854 

.9333 

.0141 

. 1391 

. 8609 

26 

44 

20 

35 

.16648 

.83352 

6.0066 

.16884 

5.9228 

1.0141 

.01395 

.98604 

25 

40 

24 

36 

. 6677 

. 3323 

5.9963 

. 6914 

.9123 

.0142 

. 1400 

. 8600 

24 

36 

28 

37 

. 6705 

. 3294 

.9860 

. 6944 

.9019 

.0142 

. 1105 

. 8595 

23 

32 

32 

38 

. 6734 

. 3266 

.9758 

. 6973 

.8915 

.0143 

. 1410 

. 8590 

22 

28 

£6 

39 

. 6763 

. 3237 

.9655 

. 7003 

.8811 

.0143 

. 1411 

. 8585 

21 

24 

40 

40 

.16791 

.83208 

5.9554 

.17033 

5.8708 

1.0114 

.01420 

.98580 

20 

20 

44 

41 

. 6820 

. 3180 

.9452 

. 7063 

.8605 

.0144 

. 1425 

. 8575 

19 

16 

48 

42 

. 6849 

. 3151 

.9351 

. 7093 

.8502 

.0145 

. 1130 

. 8570 

18 

12 

52 

43 

. 6878 

. 3122 

.9250 

. 7123 

.8400 

.0145 

. 1434 

. 8565 

17 

8 

56 

44 

. 6906 

. 3094 

.9150 

. 7153 

.8298 

.0146 

. 1439 

. 8560 

16 

4 

39 

45 

.16935 

.83065 

5.9049 

.17183 

5.8196 

1.0146 

.01444 

.98556 

15 

21 

4 

46 

. 6964 

. 3036 

.8950 

. 7213 

.8095 

.0147 

. 1449 

. 8551 

14 

56 

8 

47 

. 6992 

. 3008 

.8850 

. 7243 

.7994 

.0147 

. 1454 

. 8546 

13 

52 

12 

48 

. 7021 

. 2979 

.8751 

. 7273 

.7894 

.0148 

. 1459 

. 8541 

12 

48 

16 

49 

. 7050 

. 2950 

.8652 

. 7303 

.7793 

.0148 

. 1464 

. 85,'56 

11 

44 

20 

50 

.17078 

.82922 

5.8554 

.17333 

5.7694 

1.0149 

.01469 

.98531 

10 

40 

24 

51 

. 7107 

. 2893 

.8456 

. 7363 

.7594 

.0150 

. 1474 

. 8526 

9 

30 

28 

52 

. 7136 

. 2864 

.8358 

. 7393 

.7495 

.0150 

. 1479 

. 8521 

8 

32 

32 

53 

. 7164 

. 2836 

.8261 

. 7423 

.7396 

.0151 

. 1484 

. 8516 

7 

28 

36 

54 

. 7193 

. 2807 

.8163 

. 7453 

.7297 

.0151 

. 1489 

. 8511 

6 

21 

40 

55 

.17221 

.82778 

5.8007 

.17483 

5.7199 

1.0152 

.01494 

.98506 

5 

20 

44 

56 

. 7250 

. 2750 

.7970 

. 7513 

.7101 

.0152 

. 1499 

. 8501 

4 

16 

48 

57 

. 7279 

. 2721 

.7874 

. 7543 

.7004 

.0153 

. 1504 

. 8496 

3 

12 

52 

58 

. 7307 

. 2692 

.7778 

. 7573 

.6906 

.0153 

. 1509 

. 8491 

2 

8 

56 

59 

. 7336 

. 2661 

.7683 

. 7603 

.6809 

.0154 

. 1514 

. 8486 

1 

4 

40 

60 

. 7365 

. 2635 

.7588 

. 7633 

.6713 

.0154 

. 1519 

. 8481 

0 

20 

M. S. 

M 

Cosine. 

Vrs.Siu. 

Secant e. 

Cotang. iTangent. 

Cosec ntiVTs.Cos| 

Sine. 

M 

» V 

6 h !99° 




Natural. 



80° 

5 h 































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Natural Lines. 219 


■10° 

Natural Trigonometrical Functions 

169° 

ll h 

M 

Sine. 

Vrs. Cos. 

Cosec’nte 

Tang. 

Cotang. 

Seoante. 

Vrs. Sin 

Cosine. 

M 

M. S. 

0 

.17365 

.82636 

5.7588 

.17633 

5.6713 

1.0154 

.01519 

.98481 

60 

£0 

1 

. 7393 

. 2606 

.7493 

. 7663 

.6616 

.0155 

. 1524 

. 8476 

59 

56 

2 

. 7422 

. 2578 

.7398 

. 7693 

.6520 

.0155 

. 1529 

. 8471 

58 

52 

3 

. 7451 

. 2549 

.7304 

. 7723 

.6425 

.0156 

. 1534 

. 8465 

57 

48 

4 

. 7479 

. 2521 

.7210 

. 7753 

.6329 

.0156 

. 1539 

. 8460 

56 

44 

5 

.17508 

.82492 

5.7117 

.17783 

5.6234 

1.0157 

.01544 

.98455 

55 

40 

6 

. 7537 

. 2463 

.7023 

. 7813 

.6140 

.0157 

. 1550 

. 8450 

54 

36 

7 

. 7565 

. 2435 

.6930 

. 7843 

.6045 

.0158 

. 1555 

. 8445 

53 

32 

8 

. 7594 

. 2406 

.6838 

. 7873 

.5951 

.0158 

. 1560 

. 8440 

52 

28 

9 

. 7622 

. 2377 

.6745 

. 7903 

.5857 

.0159 

. 1565 

. 8435 

51 

24 

10 

.17651 

.82349 

5.6653 

.17933 

5.5764 

1.0159 

.01570 

.98430 

50 

20 

11 

. 7680 

. 2320 

.6561 

. 7963 

.5670 

.0160 

. 1575 

. 8425 

49 

16 

12 

. 7708 

. 2291 

.6470 

. 7993 

.5578 

.0160 

. 1580 

. 8419 

48 

12 

13 

. 7737 

. 2263 

.6379 

. 8023 

.5485 

.0161 

. 1585 

. 8414 

47 

8 

14 

. 7766 

. 2234 

.6288 

. 8053 

.5393 

.0162 

. 1591 

. 8409 

46 

4 

15 

.17794 

.82206 

5.6197 

.18083 

5.5301 

1.0162 

.01596 

.98404 

45 

19 

16 

. 7823 

. 2177 

.6107 

. 8113 

.5209 

.0163 

. 1601 

. 8399 

44 

56 

17 

. 7852 

. 2148 

.6017 

. 8143 

.5117 

.0163 

. 1606 

. 8394 

43 

52 

18 

. 7880 

. 2120 

.5928 

. 8173 

.5026 

.0164 

. 1611 

. 8388 

42 

48 

19 

. 7909 

. 2091 

.5838 

. 8203 

.4936 

.0164 

. 1CL7 

. 8383 

41 

44 

20 

.17937 

.82062 

5.5749 

.18233 

5.4845 

1.0165 

.01622 

.98378 

40 

40 

21 

. 7966 

. 2034 

.5660 

. 8263 

.4755 

.0165 

. 1627 

. 8373 

39 

36 

22 

. 7995 

. 2005 

.5572 

. 8293 

.4665 

.0166 

. 1632 

. 8368 

38 

32 

23 

. 8023 

. 1977 

.5484 

. 8323 

. .4575 

.0166 

. 1638 

. 8362 

37 

28 

24 

. 8052 

. 1948 

.5396 

. 8353 

.4486 

.0167 

. 1643 

. 8357 

36 

24 

25 

.18080 

.81919 

5.5308 

.18383 

5.4396 

1.0167 

.01648 

.98352 

35 

20 

26 

. 8109 

. 1891 

.5221 

. 8413 

.4308 

.0168 

. 1653 

. 8347 

34 

16 

27 

. 8138 

. 1862 

.5134 

. 8444 

.4219 

.0169 

. 1659 

. 8341 

33 

12 

28 

. 8166 

. 1834 

.5047 

. 8474 

.4131 

.0169 

. 1664 

. 8336 

32 

8 

29 

. 8195 

. 1805 

.4960 

. 8504 

.4043 

.0170 

. 1669 

. 8331 

31 

4 

30. 

.18223 

.81776 

5.4874 

.18534 

5.3955 

1.0170 

.01674 

.98325 

30 

18 

31 

. 8252 

. 1748 

.4788 

. 8564 

.3868 

.0171 

. 16 SO 

. 8320 

29 

56 

32 

. 8281 

; 1719 

.4702 

. 8594 

.3780 

.0171 

. 1685 

. 8315 

28 

52 

33 

. 8309 

. 1691 

.4617 

. 8624 

.3694 

.0172 

. 1690 

. 8309 

27 

48 

34 

. 8838 

. 1662 

.4532 

. 8654 

.3607 

.0172 

. 1696 

. 8304 

26 

44 

35 

.18866 

.81633 

5.4447 

.18684 

5.3521 

1.0173 

.01701 

.98299 

25 

40 

36 

. 8395 

. 1605 

.4362 

. 8714 

.3434 

.0174 

. 1706 

. 8293 

24 

36 

37 

. 8424 

. 1576 

.4278 

. 8745 

.3349 

.0174 

. 1712 

. 8288 

23 

32 

38 

. 8452 

. 1548 

.4194 

. 8775 

.3263 

.0175 

. 1717 

. 8283 

22 

28 

39 

. 8481 

. 1519 

.4110 

. 8805 

.3178 

.0175 

. 1722 

. 8277 

21 

24 

40 

.18509 

.81490 

5.4026 

.18835 

5.3093 

1.0176 

.01728 

.98272 

20 

20 

41 

. 8538 

. 1462 

.3943 

. 8865 

.3008 

.0176 

. 1733 

. 8267 

19 

1 G 

42 

. 8567 

. 1433 

.3860 

. 8895 

.2923 

.0177 

. 1739 

. 8261 

18 

12 

43 

. 8595 

. 1405 

.3777 

. 8925 

.2839 

.0177 

. 1744 

. 8256 

17 

8 

44 

. 8624 

. 1376 

.3695 

. 8955 

.2755 

.0178 

. 1749 

. 8250 

16 

4 

45 

.18652 

.81348 

5.3612 

.18985 

5.2671 

1.0179 

.01755 

.98245 

15 

17 

46 

. 8681 

. 1319 

.3530 

. 9016 

.2588 

.0179 

. 1760 

. 8240 

14 

56 

47 

. 8709 

. 1290 

.3449 

. 9046 

.2505 

.0180 

. 1766 

. 8234 

13 

52 

48 

. 8738 

. 1262 

.3367 

. 9076 

.2422 

.0180 

. 1771 

. 8229 

12 

48 

49 

. 8767 

. 1233 

.3286 

. 9106 

.2339 

.0181 

. 1777 

. 8223 

11 

44 

50 

.18795 

.81205 

5.3205 • 

.19136 

5.2257 

1.0181 

.01782 

.98218 

10 

40 

51 

. 8824 

. 1176 

.3124 

. 9166 

.2174 

.0182 

. 1788 

. 8212 

9 

36 

52 

. 8852 

. 1147 

.3044 

. 9197 

.2092 

.0182 

. 1793 

. 820*7 

8 

32 

53 

. 8881 

. 1119 

.2963 

. 9227 

.2011 

.0183 

. 1799 

. 8201 

7 

28 

54 

. 8909 

. 1090 

.2883 

. 9257 

.1929 

.0184 

. 1804 

. 8196 

6 

24 

55 

.18938 

.81062 

5.2803 

.19287 

5.1848 

1.0184 

.01810 

.98190 

5 

20 

56 

. 8967 

. 1033 

.2724 

. 9317 

.1767 

.0185 

. 18 L6 

. 8185 

4 

16 

57 

. 8995 

. 1005 

.2645 

. 9347 

.1686 

.0185 

. 1821 

. 8179 

3 

12 

58 

. 9024 

. 0976 

.2566 

. 9378 

.1606 

.0186 

. 1826 

. 8174 

2 

8 

59 

. 9U52 

. 0948 

.2487 

. 9408 

.1525 

.0186 

. 1832 

. 8168 

1 

4 

60 

. 9081 

. 0919 

.2408 

. 9438 

.1445 

.0187 

. 1837 

. 8163 

0 

1 G 

M 

Cosine. 

Yrs.SinJ Secniute. 

Cot:mg.| 

Tangent. 

Coseu’ut 

Yrs.Cos 

Sine. 

M 

M.S. 

100 ‘ 

3 



Natural. 




79° 

5 h 























220 


Natural Lines. 


o h 

ll c 

Natural Trigonometrical Functions. 

168° 

ll h 

AT. S. 

M 

Sine. 

Vrs.Cos. 

JCosee'nte 

Tang. 

Cotang. 

Secante 

Yrs. Sin 

Cosine. 

M 

M.S. 

44 

0 

.19081 

.80919 

5.2408 

.19438 

5.1445 

1.0187 

.01837 

.98163 

60 

16 

4 

1 

. 9109 

. 0890 

.2330 

. 9468 

.1366 

.0188 

. 1843 

. 8157 

59 

56 

8 

2 

. 9138 

. 0862 

.2252 

. 9498 

.1286 

.0188 

. 1848 

. 8152 

58 

52 

12 

3 

. 9166 

. 0833 

.2174 

. 9529 

.1207 

.0189 

. 1854 

. 8146 

57 

48 

16 

4 

. 9195 

. 0805 

.2097 

. 9559 

.1128 

.0189 

. 1859 

. 8140 

56 

44 

20 

5 

.19224 

.80776 

5.2019 

.19589 

5.1049 

1.0190 

.01865 

.98135 

55 

40 

24 

6 

. 9252 

. 0748 

.1942 

. 9619 

.0970 

.0191 

. 1871 

. 8121 

54 

36 

28 

7 

. 9281 

. 0719 

.1865 

. 9649 

.0892 

.0191 

. 1876 

. 8124 

53 

32 

32 

8 

. 9309 

. 0691 

.1788 

. 9680 

.0814 

.0192 

. 1882 

. 8118 

52 

28 

36 

9 

. 9338 

. 0662 

.1712 

. 9710 

.0736 

.0192 

. 1887 

. 8112 

51 

24 

40 

10 

.19366 

.80634 

5.1636 

.19740 

5.0658 

1.0193 

.01893 

.98107 

50 

20 

44 

11 

. 9395 

. 0605 

.1560 

. 9770 

.0581 

.0193 

. 1899 

. 8101 

49 

16 

48 

12 

. 9423 

. 0576 

.1484 

. 9800 

.0504 

.0194 

. 1904 

. 8095 

48 

12 

52 

13 

. 9452 

. 0548 

.1409 

. 9831 

.0427 

.0195 

. 1910 

. 8090 

47 

8 

56 

14 

. 9480 

. 0519 

.1333 

. 9861 

.0350 

.0195 

. 1916 

. 8084 

46 

4 

45 

15 

.19509 

.80491 

5.1258 

.19891 

5.0273 

1.0196 

.01921 

.98078 

45 

15 

4 

16 

. 9537 

. 0462 

.1183 

. 9921 

.0197 

.0196 

. 1927 

. 8073 

44 

56 

8 

17 

. 9566 

. 0434 

.1109 

. 9952 

.0121 

.0197 

. 1933 

. 8067 

43 

52 

12 

18 

. 9595 

. 0405 

.1034 

. 9982 

.0045 

.0198 

. 1938 

. 8061 

42 

48 

16 

19 

. 9623 

. 0377 

.0960 

.20012 

4.9969 

.0198 

. 1944 

. 8056 

41 

44 

20 

20 

.19652 

.80348 

5.0886 

.20042 

4.9894 

1.0199 

.01950 

.98050 

40 

40 

24 

21 

. 9680 

. 0320 

.0812 

. 0073 

.9819 

.0199 

. 1956 

. 8044 

39 

3G 

28 

22 

. 9709 

. 0291 

.0739 

. 0103 

.9744 

.0200 • 

. 1961 

. 8039 

38 

32 

32 

23 

. 9737 

. 0263 

.0666 

. 0133 

.9669 

.0201 

. 1967 

. 8033 

37 

28 

36 

24 

. 9766 

. 0234 

.0593 

. 0163 

.9594 

.0201 

. 1973 

. 8027 

36 

24 

40 

25 

.19794 

.80206 

5.0520 

.20194 

4.9520 

1.0202 

.01979 

.98021 

35 

20 

44 

26 

. 9823 

. 0177 

.0447 

. 0224 

.9446 

.0202 

. 1984 

. 8016 

34 

16 

48 

27 

. 9851 

. 0149 

.0375 

. 0254 

.9372 

.0203 

. 1990 

. 8010 

33 

12 

52 

28 

. 9880 

. 0120 

.0302 

. 0285 

.9298 

.0204 

. 1996 

. 8004 

32 

8 

56 

29 

. 9908 

. 0092 

.0230 

. 0315 

.9225 

.0204 

. 2002 

. 7998 

31 

4 

40 

30 

.19937 

.80063 

5.0158 

.20345 

4.9151 

1.0205 

.02007 

.97992 

30 

IT 

4 

31 

. 9965 

. 0035 

.0087 

. 0375 

.9078 

.0205 

. 2013 

. 7987 

29 

56 

8 

32 

. 9994 

. 0006 

.0015 

. 0406 

.9006 

.0206 

. 2019 

. 7981 

28 

52 

12 

33 

.20022 

.79978 

4.9944 

. 0436 

.8933 

.0207 

. 2025 

. 7975 

27 

48 

16 

34 

.20051 

.79949 

4.9873 

. 0466 

.8860 

.0207 

. 2031 

. 7969 

26 

44 

20 

35 

.20079 

.79921 

4.9802 

.20497 

4.8788 

1.0208 

:02037 

.97963 

25 

40 

24 

36 

. 0108 

. 9892 

.9732 

. 0527 

.8716 

.0208 

. 2042 

. 7957 

24 

36 

28 

37 

. 0136 

. 9863 

.9661 

. 0557 

.8644 

.0209 

. 2048 

. 7952 

23 

32 

32 

38 

. 0165 

. 9835 

.9591 

. 0588 

.8573 

.0210 

. 2054 

. 7946 

22 

28 

36 

39 

. 0193 

. 9807 

.9521 

. 0618 

.8501 

.0210 

. 2060 

. 7940 

21 

24 

40 

40 

.20222 

.79778 

4.9452 

.20048 

4.8430 

1.0211 

.02066 

.97934 

20 

20 

44 

41 

. 0250 

. 9750 

.9382 

. 0679 

.8359 

.0211 

. 2(J72 

. 7928 

19 

16 

48 

42 

. 0279 

. 9721 

.9313 

. 0709 

.S2S8 

.0212 

. 2078 

. 7922 

18 

12 

52 

43 

. 0307 

. 9693 

.9243 

. 0739 

.8217 

.0213 

. 2084 

. 7916 

17 

8 

56 

44 

. 0336 

. 9664 

.9175 

. 0770 

.8147 

.0213 

. 2089 

. 7910 

16 

4 

47 

45 

.20564 

.79636 

4.9106 

.20800 

4.8077 

1.0214 

.02095 

.97964 

15 

13 

4 

46 

. 0393 

. 9607 

.9037 

. 0830 

.8007 

.0215 

. 2101 

. 7899 

14 

56 

8 

47 

. 0421 

. 9579 

.8969 

. 0861 

.7937 

.0215 

. 2107 

. 7893 

13 

52 

12 

48 

. 0450 

. 9550 

.8901 

. 0891 

.7867 

.0216 

. 2113 

. 7887 

12 

48 

16 

49 

. 0478 

. 9522 

.8833 

. 0921 

.7798 

.0216 

. 2119 

. 7881 

11 

44 

20 

50 

.20506 

.79493 

4.8765 

.20952 

4.7728 

1.0217 

.02125 

.97875 

10 

40 

24 

51 

. 0535 

. 9465 

.8697 

. 0982 

.7659 

.0218 

. 2131 

. 7869 

9 

36 

28 

52 

.'0563 

. 9436 

.8630 

. 1012 

.7591 

.0218 

. 2137 

. 7863 

8 

32 

32 

53 

. 0592 

. 9408 

.8563 

. 1043 

.7522 

.0219 

. 2143 

. 7857 

7 

28 

36 

54 

. 0620 

. 9379 

.8496 

. 1073 

.7453 

.0220 

. 2149 

. 7851 

6 

24 

40 

55 

.20649 

.79351 

4.8429 

.21104 

4.7385 

1.0220 

.02155 

.97845 

5 

20 

44 

56 

. 06 17 

. 9323 

.8362 

. 1134 

.7317 

.0221 

. 2161 

. 7839 

4 

16 

48 

57 

. 0706 

. 9294 

.8296 

. 1164 

.7249 

.022 L 

. 2167 

. 7833 

3 

12 

52 

58 

. 0734 

. 9266 

.8229 

. 1195 

.7181 

.0222 

. 2173 

. 7827 

2 

8 

56 

59 

. 0763 

. 9237 

.8163 

. 1225 

.7114 

.0223 

. 2179 

. 7821 

1 

4 

48 

60 

. 0791 

. 9209 

.8097 

. 1256 

.7046 

.0223 

. 2185 

. 7815 

0 

12 

M. S. 

M 

Cosine. 

Vrs.Sin. 

Secante. 

Cotuug.jTaugeut. 

Cosec'nt 

Vrs.Cos 

Sine. 

M 

M.S. 

6 h 

101 ° 



Natural. 



78° 

5 h 



























Natural Lines. 


221 


o h 

12° 

Natural Trigonometrical Functions. 

167° 

ll h 

M. S. 

M 

Sine. 

Vrs.Cos .jjCosec'nte 

Tang. 

Cotang. 

Secante 

i Vrs.Sin 

Cosine. 

M 

M.S. 

4-S 

0 

.20791 

.79209 

4.8097 

.21250 

4.7046 

1.0223 

.02185 

.97815 

60 

14 

4 

1 

. 0820 

. 9180 

.8032 

. 1286 

.6979 

.0224 

. 219 L 

. 7809 

59 

56 

8 

2 

. 0848 

. 9152 

.7966 

. 1316 

.6912 

.0225 

. 2197 

.• 7803 

5S 

52 

12 

3 

. 0876 

. 9123 

.7901 

. 1347 

.6845 

.0225 

. 2203 

. 7806 

57 

48 

1(3 

4 

. 0905 

. 9105 

.7835 

. 1377 

.6778 

.0226 

j. 2209 

. 7790 

56 

44 

20 

5 

.20933 

.79066 

4.7770 

.21408 

4.6712 

1.0226 

.02215 

.97784 

55 

40 

24 

6 

. 0962 

. 9038 

.7706 

. 1438 

.6646 

.0227 

|. 2222 

. 7778 

54 

36 

28 

7 

. 0990 

. 9010 

.7641 

. 1168 

.6580 

.0228 

|. 2228 

. 7772 

53 

32 

32 

8 

. 1019 

. 8981 

.7576 

. 1499 

.6514 

.0228 

. 2234 

. 7706 

52 

28 

36 

9 

. 1047 

. 8953 

.7512 

. 1529 

.6448 

.0229 

. 2240 

. 7760 

51 

24 

40 

10 

.21070 

.78924 

4.7448 

.21560 

4.6382 

1.0230 

.02246 

.97754 

50 

20 

44 

n 

. 1104 

. 8896 

.7384 

. 1590 

.6317 

.0230 

. 2252 

. 7748 

19 

16 

48 

12 

. 1132 

. 8867 

.7320 

. 1621 

.6252 

.0231 

. 2258 

. 7741 

48 

12 

62 

13 

. 1161 

. 8839 

.7257 

. 1651 

.6187 

.0232 

. 2264 

. 7735 

47 

8 

56 

14 

. 1189 

. 8811 

.7193 

. 1682 

.6122 

.0232 

. 2271 

. 7729 

46 

4 

T9 

15 

.21218 

.78782 

4.7130 

.21712 

4.6057 

1.0233 

.02277 

.97723 

45 

11 

4 

16 

. 1246 

. 8754 

.7067 

. 1742 

.5993 

.0234 

. 2283 

. 7717 

44 

56 

s 

17 

. 1275 

. 8725 

.70(4 

. 1773 

.5928 

.0234 

. 2289 

. 7711 

43 

52 

12 

18 

. 1303 

. 8697 

.6942 

. 1803 

.5864 

.0235 

. 2295 

. 7704 

42 

48 

16 

19 

. 1331 

. 8668 

.6879 

. 1834 

.5800 

.0235 

. 2302 

. 7698 

41 

44 

20 

20 

.21360 

.78640 

4.6817 

.21864 

4.5136 

1.0236 

.02308 

.97692 

40 

40 

24 

21 

. 1388 

. 8612 

.6754 

. 1895 

.5673 

.0237 

. 2314 

. 7686 

39 

36 

28 

22 

. 1417 

. 8583 

.6692 

. 1925 

.5609 

.0237 

. 2320 

. 7680 

38 

32 . 

32 

23 

. 1145 

. 8555 

.663 L 

. 1956 

.5546 

.0238 

. 2326 

. 7673 

37 

28 

36 

24 

. 1473 

. 8526 

.65* 9 

. 1986 

.5483 

.0239 

. 2333 

. 7667 

36 

24 

40 

25 

.21502 

.78508 

4.6507 

.22017 

4.5420 

1.0239 

.02339 

.97661 

35 

20 

44 

26 

. 1530 

. 8470 

.6446 

. 2047 

.5357 

.0240 

. 2345 

. 7655 

34 

16 

48 

27 

. 1559 

. 8441 

.6385 

. 2078 

.5294 

.0241 

. 2351 

. 7618 

33 

12 

52 

28 

. 1587 

. 8413 

.6324 

. 2108 

.5232 

.0241 

. 2358 

. 7642 

32 

8 

56 

29 

. 1615 

. 8384 

.6263 

. 2139 

.5169 

.0242 

. 2364 

. 7636 

31 

4 

50 

30 

.21644 

.78356 

4.6202 

.22169 

4.5107 

1.0243 

.02370 

.97630 

30 

10 

4 

31 

. 1672 

. 8328 

.6142 

. 2200 

.50 i 5 

.0243 

. 2377 

. 7623 

29 

56 

8 

32 

. 1701 

. 8299 

.6081 

. 2230 

.4983 

.0244 

. 2383 

. 7617 

28 

52 

12 

33 

. 1729 

. 8271 

.6021 

. 2261 

.4921 

.0245 

. 238 9 

. 7611 

27 

48 

16 

34 

. 1757 

. 8242 

.5961 

. 2291 

.4860 

.0245 

. 2396 

. 7604 

26 

44 

20 

35 

.21786 

.78214 

4.5901 

.22322 

4.4799 

1.0246 

.02402 

.97598 

25 

40 

24 

36 

. 1814 

. 8186 

.5841 

. 2353 

.4737 

.0217 

. 2408 

. 7 592 

24 

36 

2S 

37 

. 1813 

. 8154 

.5782 

. 2383 

.4676 

.0247 

. 2415 

. 7585 

23 

32 

32 

38 

. 1871 

. 8129 

.5722 

. 2414 

.4615 

.0248 

. 2421 

. 7579 

22 

28 

3G 

39 

. 1899 

. 8100 

.5663 

. 2444 

.4555 

.0249 

. 2427 

. 7573 

21 

21 

40 

40 

.21928 

.78072 

4.5604 

.22475 

4.4494 

1.0249 

.02434 

.97566 

20 

20 

44 

41 

. 1956 

. 8043 

.5515 

. 2505 

.4434 

.0250 

. 2440 

. 7560 

19 

16 

48 

42 

. 1985 

. 8015 

.5486 

. 2536 

.4373 

.0251 

. 2446 

. 7553 

18 

12 

52 

43 

. 2013 

. 7987 

.5428 

. 2566 

.4313 

.0251 

. 2453 

. 7547 

17 

8 

56 

44 

. 2041 

. 7959 

.5369 

. 2597 

.4253 

.0252 

i 2459 

. 7541 

16 

4 

51 

45 

.22070 

.77930 

4.5311 

.22628 

4.4194 

1.0253 

.02466 

.97534 

15 

9 

4 

46 

. 2098 

. 7902 

.5253 

. 2658 

.4131 

.0253 

. 2472 

. 7528 

14 

56 

S 

47 

. 2126 

. 7873 

.5195 

. 2689 

.4074 

.0254 

. 2479 

. 7521 

13 

52 

12 

48 

. 2155 

. 7845 

.5137 

. 2719 

.4015 

.0255 

. 2185 

. 7515 

12 

48 

16 

49 

. 2183 

. 7817 

.5079 

. 2750 

.3956 

.0255 

. 2491 

. 7508 

11 

44 

20 

50 

.22211 

.77788 

4.5021 

.22781 

4.3897 

1.0256 

.02498 

.97502 

10 

40 

24 

51 

. 2240 

. 7760 

.4964 

. 2811 

.3838 

.0257 

. 2504 

. 7495 

9 

36 

28 

52 

. 2268 

. 7732 

.4907 

. 2842 

.3779 

.0257 

. 2511 

. 7489 

8 

32 

32 

53 

. 2297 

. 7703 

.4850 

. 2872 

.3721 

.0258 

. 2517 

. 7483 

7 

28 

36 

54 

. 2325 

. 7675 

.4793 

. 2903 

.3662 

.0259 

. 2521 

. 7476 

6 

24 

40 

55 

.22353 

.77647 

4.4736 

.22934 

4.3604 

1.0260 

.02530 

.97470 

5 

20 

44 

56 

. 2382 

. 7618 

.4679 

. 2964 

.3546 

.0260 

. 2537 

. 7463 

4 

16 

48 

57 

. 2410 

. 7590 1 

.4623 

. 2995 

.3488 

.0261 1 

. 2543 

. 7457 

3 

12 

52 

58 

. 2438 

. 7561 | 

.4566 

. 3025 

.3430 

.0262 « 

. 2550 

. 7450 

2 

8 

5(5 

59 

. 2467 

. 7533 | 

.4510 

. 3056 

.3372 

.0262 

. 2556 

. 7443 

1 

4 

5?3 

60 

. 2495 

. 7505 

.4454 

. 3087 | 

.3315 

.0263 

. 2563 

. 7437 

0 

8 

M. S. 

M 

Cosine. 

Vrs.Sin.! 

Senaute. 

Cotaug.j 

Tangent. 

Cosec'nt! 

Vrs.Cos 

Sine. 

M 

M.S 

6 h 

102° 



Natural. 



77° 

5 b 

































222 


Natural Lines. 


o h 

13 c 

Natural Trigonometrical Functions 

166° 

jll h 

M. S. 

M 

Sine. 

Vrs. Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

| Cosine. 

M 

M.S. 

5 i 

0 

.22495 

.77505 

4.4454 

.23087 

4.3315 

1.0263 

.02563 

1 .97437 

60 

8 

4 

1 

. 2523 

. 7476 

.4398 

. 3117 

.3257 

.0261 

. 2569 

. 7430 

59 

56 

8 

2 

. 2552 

. 7448 

.4342 

. 3148 

.3200 

.0264 

. 2576 

. 7424 

58 

52 

12 

3 

. 2580 

. 7420 

.4287 

. 3179 

.3143 

.0265 

. 2583 

. 7417 

57 

4S 

16 

4 

. 2608 

. 7391 

.4231 

. 3209 

.3086 

.0266 

. 2589 

. 7411 

56 

44 

20 

5 

.22637 

.77363 

4.4176 

.23240 

4.3029 

1.0266 

.02596 

.97404 

5o 

-40 

24 

6 

. 2665 

. 7335 

.4121 

. 3270 

.2972 

.0267 

. 2602 

. 7398 

54 

36 

28 

7 

. 2693 

. 7306 

.4065 

. 3301 

.2916 

.0268 

. 2609 

. 7391 

53 

32 

32 

8 

. 2722 

. 7278 

.4011 

. 3332 

.2859 

.0268 

. 2616 

. 73S4 

52 

28 

36 

9 

. 2750 

. 7250 

.3956 

. 3363 

.2803 

.0269 

. 2622 

. 7378 

51 

24 

40 

10 

.22778 

.77221 

4.3901 

.23393 

4.2747 

1.0270 

.02629 

.97371 

50 

20 

44 

11 

. 2807 

. 7193 

.3847 

. 3424 

.2691 

.0271 

. 2635 

. 7364 

49 

16 

48 

12 

. 2835 

. 7165 

.3792 

. 34 ') 5 

.2635 

.0271 

. 2642 

. 7358 

48 

12 

52 

13 

. 2863 

. 7136 

.3738 

. 3485 

.2579 

.0272 

. 2649 

. 7351 

47 

8 

56 

14 

. 2892 

. 7108 

.3684 

. 3516 

.2524 

.0273 

. 2655 

. 7344 

46 

4 

53 

15 

.22920 

.77080 

4.3630 

.23547 

4.2468 

1.0273 

.02662 

.97338 

45 

7 

4 

16 

. 2948 

. 7052 

.3576 

. 3577 

.2413 

.0274 

. 2669 

. 7331 

44 

56 

8 

17 

. 2977 

. 7023 

.3522 

. 3608 

.2358 

.0275 

. 2675 

. 7324 

43 

52 

12 

18 

. 3005 

. 6995 

.3469 

. 3639 

.2303 

.0276 

. 2682 

. 7318 

42 

48 

16 

19 

. 3033 

. 6967 

.3415 

. 3670 

.2248 

.0276 

. 2689 

. 7311 

41 

44 

20 

20 

.23061 

.76938 

4.3362 

.23700 

4.2193 

1.0277 

.02695 

.97304 

40 

4 > 

24 

21 

. 3090 

. 6910 

.3309 

. 3731 

.2139 

.0278 

. 2702 

. 7298 

39 

36 

► 28 

22 

. 3118 

. 6882 

.3256 

. 3762 

.2084 

.0278 

. 2709 

. 7291 

38 

32 

32 

23 

. 3146 

. 6853 

.3203 

. 3793 

.2030 

.0279 

. 2716 

. 7284 

37 

28 

36 

24 

. 3175 

. 6825 

.3150 

. 3823 

.1976 

.0280 

. 2722 

. 7277 

36 

24 

40 

25 

.23203 

.76797 

4.3098 

.23854 

4.1921 

1.0280 

.02729 

.97271 

35 

20 

44 

26 

. 3231 

. 6769 

.3045 

. 3885 

.1867 

.0281 

. it736 

. 7264 

34 

16 

48 

27 

. 3260 

. 6740 

.2993 

. 3916 

.1814 

.0282 

. 2743 

. 7257 

33 

12 

52 

28 

. 3288 

. 6712 

.2941 

. 3946 

.1760 

.0283 

. 2749 

. 7250 

32 

8 

50 

29 

. 3316 

. 6684 

.2888 

. 3977 

.1706 

.0283 

. 2756 

. 7244 

31 

4 

54 

30 

.23344 

.76655 

4.2836 

.24008 

4.1653 

1.0284 

.02763 

.97237 

30 

6 

4 

31 

. 3373 

. 6627 

.2785 

. 4039 

.1600 

.0285 

. 2770 

. 7230 

29 

56 

8 

32 

. 3401 

. 6599 

.2733 

. 4069 

.1546 

.0285 

. 2777 

. 7223 

28 

52 

12 

33 

. 3429 

. 6571 

.2681 

. 4100 

.1493 

.0280 

. 2783 

. 7216 

27 

48 

16 

34 

. 3458 

. 6542 

.2630 

. 4131 

.1440 

.0287 

. 2790 

. 7210 

26 

44 

20 

35 

.23486 

.76514 

4.2579 

.24102 

4.1388 

1.0288 

.02797 

.97203 

25 

40 

24 

36 

. 3514 

. 6486 

.2527 

. 4192 

.1335 

.0288 

. 2804 

. 7196 

24 

36 

28 

37 

. 3542 

. 6457 

.2476 

. 4223 

.1282 

.0289 

. 2811 

. 7189 

23 

32 

32 

38 

. 3571 

. 6429 

.2425 

. 4254 

.1230 

.0290 

. 2818 

. 7182 

22 

28 

36 

39 

. 3599 

. 6401 

.2375 

. 4285 

.1178 

.0291 

. 2824 

. 7175 

21 

24 

40 

40 

.23627 

.76373 

4.2324 

.24316 

4.1126 

1.0291 

.02831 

.97169 

20 

20 

44 

41 

. 3655 

. 6344 

.2273 

. 4346 

.1073 

.0292 

. 2838 

. 7162 

19 

16 

48 

42 

. 3684 

. 6316 

.2223 

. 4377 

.1022 

.0293 

. 2845 

. 7155 

18 

12 

52 

43 

. 3712 

. 6288 

.2173 

. 4408 

.0970 

.0293 

. 2852 

. 7148 

17 

8 

56 

44 

. 3740 

. 6260 

.2122 

. 4439 

.0918 

.0294 

. 2859 

. 7141 

16 

4 

55 

45 

.23768 

.76231 

4.2072 

.24470 

4.0867 

1.0295 

.02866 

.97134 

15 

5 

4 

46 

. 3797 

. 6203 

.2022 

. 4501 

.0815 

.0296 

. 2873 

. 7127 

14 

56 

8 

47 

. 3825 

. 6175 

.1972 

. 4531 

.0764 

.0296 

. 2880 

. 7120 

13 

52 

12 

48 

. 3853 

. 6147 

.1923 

. 4562 

.0713 

.0297 

. 2886 

. 7113 

12 

48 

16 

49 

. 3881 

. 6118 

.1873 

. 4593 

.0662 

.0298 

. 2893 

. 7106 

11 

44 

20 

50 

.23910 

.76090 

4.1824 

.24624 

4.0611 

1.0299 

.02900 

.97099 

10 

40 

24 

51 

. 3938 

. 6062 

.1774 

. 4655 

.0560 

.0299 

. 2907 

. 7092 

9 

36 

28 

52 

. 3966 

. 6034 

.1725 

. 4686 

.0509 

.0300 

. 2914 

. 7086 

8 

32 

32 

53 

. 3994 

. 6005 

.1676 

. 4717 

.0458 

.0301 

. 2921 

. 7079 

7 

28 

36 

54 

. 4023 

. 5977 

.1627 

. 4747 

.0408 

.0302 

. 2928 

. 7072 

6 

24 

40 

55 

.24051 

.75949 

4.1578 

.24778 

4.0358 

1.0302 

.02935 

.97065 

5 

20 

44 

56 

. 4079 

. 5921 

.1529 

. 4809 

.0307 

.0303 

. 2942 

. 7058 

4 

16 

48 

57 

. 4107 

. 5892 

.1481 

. 4840 

.0257 

.0304 

. 2949 

. 7051 

3 

12 

52 

5S 

. 4136 

. 5864 

.1432 

. 4^71 

.0207 

.0305 

. 2956 

. 7044 

2 

8 

56 

59 

. 4164 

. 5836 

.1384 

. 4902 

.0157 

.0305 

. 2963 

. 7037 

1 

4 

50 

60 

. 4192 

. 5808 

.1336 

. 4933 

.0108 

.0306 

. 2970 

. 7029 

0 

4 

M.S. 

6 h 

i 

M 

103' 

Cosiue. 

D 

Vrs.Sin. 

Secants. 

Cotaug.iTaugent. 

Natural. 

Cosec’ntlVrs.Cos 

Sine. 

M 

76° 

M.S. 

5 h 





























Natural Lines. 


223 


O h 

14° latural Trig 

onometrical Functions. 

165° 

ll h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante 

' Vrs.Sin 

Cosine. 

I M 

M.S. 


0 

.24192 

.75808 

4.1336 

.24933 

4.0108 

1.0306 

.02970 

.97029 

60 

4 

4 

1 

. 4220 

. 5779 

.1287 

. 4964 

.0058 

.0307 

. 2977 

. 7022 

59 

56 

8 

2 

. 4249 

. 5751 

.1239 

. 4995 

.0009 

.0308 

. 2984 

. 7015 

58 

52 

12 

3 

. 4277 

. 5723 

.1191 

. 5025 

.9959 

.0308 

. 2991 

. 7008 

57 

48 

1 G 

4 

. 4305 

. 5695 

.1141 

. 5056 

3.9910 

.0309 

. 2999 

. 7001 

56 

44 

20 

5 

.24333 

.75667 

4.1096 

.25087 

3.9861 

1.0310 

.03006 

.96994 

55 

40 

24 

6 

. 4361 

. 5638 

.1048 

. 5118 

.9812 

.0311 

. 3013 

. 6987 

54 

36 

28 

7 

. 4390 

. 5610 

.1001 

. 5149 

.9763 

.0311 

. 3020 

. 6980 

53 

32 

32 

8 

. 4418 

. 5582 

• .0953 

. 5180 

.9711 

.0312 

. 3027 

. 6973 

52 

28 

30 

9 

. 4446 

. 5554 

.0906 

. 5211 

.9665 

.0313 

. 3034 

. 6966 

51 

24 

40 

10 

.24474 

.75526 

4.0859 

.25242 

3.9616 

1.0314 

.03041 

.96959 

50 

20 

‘FT- 

11 

. 4502 

. 5497 

.0812 

. 5273 

.9568 

.0314 

. 3048 

. 6952 

49 

16 

48 

12 

. 4531 

. 5469 

.0765 

. 5304 

.9520 

.0315 

. 3055 

. 6944 

48 

12 

52 

13 

. 4559 

. 5441 

.0718 

. 5335 

.9471 

.0316 

. 3063 

. 6937 

47 

8 

5G 

14 

. 4587 

. 5413 

.0672 

. 5366 

.9423 

.0317 

. 3070 

. 6930 

46 

4 

57 

15 

.24615 

.75385 

4.0625 

.25397 

3.9375 

1.0317 

.03077 

.96923 

45 

3 

4 

16 

. 4643 

. 5356 

.0579 

. 5428 

.9327 

.0318 

. 3084 

. 6916 

44 

56 

s 

17 

. 4672 

. 5328 

.0532 

. 5459 

.9279 

.0319 

. 3091 

. 6909 

43 

52 

12 

18 

. 4700 

. 5300 

.0486 

. 5490 

.9231 

.0320 

. 3098 

. 6901 

42 

48 

16 

19 

. 4728 

. 5272 

.0440 

. 5521 

.9184 

.0320 

. 3106 

. 6894 

41 

44 

20 

20 

.24756 

.75244 

4.0394 

.25552 

3.9136 

1.0321 

.03113 

.66887 

40 

40 

24 

21 

. 4784 

. 5215 

.0348 

. 5583 

.9089 

.0322 

. 3120 

. 6880 

39 

36 

28 

22 

. 4813 

. 5187 

.0302 

•. 5614 

.9042 

.0323 

. 3127 

. 6873 

38 

32 

32 

23 

. 4841 

. 5159 

.0256 

. 5645 

.8994 

.0323 

. 3134 

. 6865 

37 

28 

36 

24 

. 4869 

. 5131 

.0211 

. 5676 

.8947 

.0324 

. 3142 

. 6858 

36 

24 

40 

25 

.24897 

.75103 

4.0165 

.25707 

3.S900 

1.0325 

.03149 

.96851 

35 

20 

44 

26 

. 4925 

. 5075 

.0120 

. 5738 

.8853 

.0326 

. 3156 

. 6844 

34 

16 

48 

27 

. 4953 

. 5046 

.0074 

. 5769 

.8807 

.0327 

. 3163 

. 6836 

33 

12 

52 

28 

. 4982 

. 5018 

.0029 

. 5800 

.8760 

.0327 

. 3171 

. 6829 

32 

8 

56 

29 

. 5010 

. 4990 

3.9984 

. 5831 

.8713 

.0328 

. 3178 

. 6822 

31 

4 

58 

30 

.25038 

.74962 

3.9939 

.25862 

3.8667 

1.0329 

.03185 

.96815 

30 

2 

4 

31 

. 5066 

. 4934 

.9894 

. 5893 

.8621 

.0330 

. 3192 

. 6807 

29 

56 

8 

32 

. 5094 

. 4906 

.9850 

. 5924 

.8574 

.0330 

. 3200 

. 6800 

28 

52 

12 

33 

. 5122 

. 4877 

.9805 

. 5955 

.8528 

.0331 

. 3207 

. 6793 

27 

48 

16 

34 

. 5151 

. 4849 

.9760 

. 5986 

.8482 

.0332 

. 3214 

. 6785 

26 

44 

. 20 

35 

.25179 

.74821 

3.9716 

.26017 

3.8436 

1.0333 

.03222 

.96778 

25 

40 

24 

36 

. 5207 

. 4793 

.9672 

. 6048 

.8390 

.0334 

. 3229 

. 6771 

24 

36 

28 

37 

. 5235 

. 4765 

.9627 

. 6079 

.8345 

.0334 

. 3236 

. 6763 

23 

32 

32 

38 

. 5263 

. 4737 

.9583 

. 6110 

.8299 

.0335 

. 3244 

. 6756 

22 

28 

c6 

39 

. 5291 

. 4709 

.9539 

. 6141 

.8254 

.0336 

. 3251 

. 6749 

21 

24 

40 

40 

.25319 

.74680 

3.9495 

.26172 

3.8208 

1.0337 

.03258 

.96741 

20 

20 

44 

41 

. 5348 

. 4652 

.9451 

. 6203 

.8163 

.0338 

. 3266 

. 6734 

19 

16 

48 

42 

. 5376 

. 4624 

.9408 

. 6234 

.8118 

.0338 

. 3273 

. 6727 

18 

12 

52 

43 

. 5404 

. 4596 

.9364 

. 6266 

.8073 

.0339 

. 3281 

. 6719 

17 

8 

56 

44 

. 5432 

. 4568 

.9320 

. 6297 

.8027 

.0340 

. 3288 

. 6712 

16 

4 

51) 

45 

.25460 

.74540 

3.9277 

.26328 

3.7983 

1.0341 

.03295 

.96704 

15 

1 

4 

46 

. 5488 

. 4512 

.9234 

. 6359 

.7938 

.0341 

. 3303 

. 6697 

14 

56 

8 

47 

. 5516 

. 4483 

.9190 

. 6390 

.7893 

.0342 

. 3310 

. 6690 

13 

52 

12 

48 

. 5544 

. 4455 

.9147 

. 6421 

.7848 

.0343 

. 3318 

. 6682 

12 

48 

16 

49 

. 5573 

. 4427 

.9104 

. 6452 

.7804 

.0344 

. 3325 

. 6675 

11 

44 

20 

50 

25601 

.74399 

3.9061 

.26483 

3.7759 

1.0345 

.03332 

.96667 

10 

40 

24 

51 

. 5629 

. 4371 

.9018 

. 6514 

.7715 

.0345 

. 3340 

. 6660 

9 

36 

28 

52 

. 5657 

. 4344 

.8976 

. 6546 

.7671 

.0346 

. 3347 

. 6652 

8 

32 

32 

53 

. 5685 

. 4315 

.8033 

. 6577 

.7627 

.0347 

. 3355 

. 6645 

7 

28 

36 

54 

. 5713 

. 4287 

.8890 

. 6608 

.7583 

.0348 

. 3362 

. 6638 

6 

24 

40 

55 

.25741 

.74259 

3.8848 

.26639 

3.7539 

1.0349 

.03370 

.96630 

5 

20 

44 

56 

. 5769 

. 4230 

.8805 

. 6670 

.7495 

.0349 

. 3377 

. 6623 

4 

16 

48 

57 

• 57 98 

. 4202 

.87 63 

. 6701 

.7451 

.0350 

. 3385 

. 6615 

3 

12 

52 

58 

. 5826 

. 4174 

.8721 

. 6732 

.7407 

.0351 

. 3392 

. 6608 

2 

8 

56 

59 

. 5S54 

. 4146 

.SC79 

. 6764 

.73(4 

.0352 

. 3400 

. 6600 

1 

4 

60 

60 

. 5882 

. 4118 

.8637 

. 6795 

.7320 

.0353 

. 3107 

. 6592 

0 

O 

M. S. 

M 

Cosine. 

Vrs.Sin. 

Secante. 1 Cotaug.iTangeni. 

Cosec’nt 

Vrs.Cos 

Sine. 

M 

M.S. 

6 b 

104 c 




Natural. 



75° 

5 b 
































224 Natural Lines. 


1 

l h 

! 15' 

3 

Natural Trigonometrical Functions 

164° 

10 h 

M.S. 

M 

Sine. 

| Vrs. Cos. 

Cosec’nte 

Tn.ng. 

Cotang. 

Secante 

Vrs. Sin 

Cosine. 

M 

M. S. 

0 

0 

.25882 

.74118 

3.8637 

.26795 

3.7320 

1.0353 

.03407 

.96592 

60 

60 

4 

i 

. 5910 

. 4090 

.8595 

. 6826 

.7277 

.0353 

. 3415 

. 6585 

59 

56 

8 

2 

. 5938 

. 4002 

.8553 

. 6857 

.7234 

.0354 

. 3422 

. 6577 

58 

52 

12 

3 

. 5966 

. 4034 

.8512 

. 6888 

.7191 

.0355 

. 3430 

. 6570 

57 

48 

16 

4 

. 5994 

. 4006 

.8470 

. 6920 

.7147 

.0356 

. 3438 

. 6562 

56 

44 

20 

5 

.26022 

.73978 

3.8428 

.26951 

3.7104 

1.0357 

.03445 

.96555 

55 

40 

24 

6 

. 6050 

. 3949 

.8387 

. 6982 

.7002 

.0358 

. 3453 

. 6547 

54 

36 

28 

7 

. 6078 

. 3921 

.8346 

. 7013 

.7019 

.0358 

. 3460 

. 6540 

53 

32 

32 

8 

. 6107 

. 3893 

.8304 

. 7044 

.6976 

.0359 

. 3468 

. 6532 

52 

28 

36 

9 

. 6135 

. 3865 

.8263 

. 7076 

.6933 

.0360 

. 3475 

. 6524 

51 

24 

40 

10 

.26163 

.73837 

3.8222 

.27107 

3.6891 

1.0361 

.03483 

.96517 

50 

20 

44 

11 

. 6191 

. 3809 

.8181 

. 7138 

.6848 

.0302 

. 3491 

. 6509 

49 

16 

48 

12 

. 6219 

. 3781 

.8140 

. 7169 

.6806 

.0302 

. 3498 

. 6502 

48 

12 

52 

13 

. 6247 

. 3753 

.8100 

. 7201 

.6764 

.0363 

. 3506 

. 6494 

47 

8 

50 

14 

. 6275 

. 3725 

.8059 

. 7232 

.6722 

.0364 

. 3514 

. 6486 

46 

4 

1 

15 

.26303 

.73697 

3.8018 

.27263 

3.6679 

1.0305 

.03521 

.96479 

45 

59 

4 

16 

. 6331 

. 3669 

.7978 

. 7294 

.6637 

.0366 

. 3529 

. 6471 

44 

56 

8 

17 

. 6359 

. 3641 

.7937 

. 7326 

.6596 

.0367 

. 3536 

. 6463 

43 

52 

12 

18 

. 6387 

. 3013 

.7897 

. 7357 

.6554 

.0307 

. 3544 

. 6456 

42 

48 

16 

19 

. 6415 

. 3585 

.7857 

. 7388 

.6512 

.0368 

. 3552 

. 6448 

41 

44 

20 

20 

.20443 

.73556 

3.7816 

.27419 

3.0470 

1.0309 

.03560 

.96440 

40 

40 

24 

21 

. 647 L 

. 3528 

.7776 

. 7451 

.6429 

.0370 

. 3567 

. 6483 

39 

36 

28 

22 

. 6499 

. 3500 

.7736 

. 7482 

.6387 

.0371 

. 3575 

. 6425 

38 

32 

32 

23 

. 6527 

. 3472 

.7697 

. 7513 

.6346 

.0371 

. 3583 

. 6417 

37 

28 

36 

24 

. 6550 

. 3444 

.7657 

. 7544 

.6305 

.0372 

. 3590 

. 6109 

36 

24 

40 

25 

.26584 

.73416 

3.7617 

.27576 

3.6263 

1.0373 

.03598 

.96402 

35 

20 

44 

26 

. 6612 

. 338S 

.7577 

. 7607 

.6222 

.0374 

. 3606 

. 6394 

34 

16 

48 

27 

. 6040 

. 3360 

.7538 

. 7638 

.6181 

.0375 

. 3614 

. 6386 

33 

12 

52 

28 

. 6668 

. 3332 

.7498 

. 7670 

.0140 

.0376 

. 3621 

. 6378 

32 

8 

56 

29 

. 6696 

. 3304 

.7459 

. 7701 

.6100 

.0376 

. 3629 

. 6371 

31 

4 

2 

30 

.26724 

.73276 

3.7420 

.27732 

3.6059 

1.0377 

.03637 

.96363 

30 

58 

4 

31 

. 6752 

. 3248 

.7380 

. 7764 

.6018 

.0378 

. 3645 

. 6355 

29 

56 

8 

32 

. 6780 

. 3220 

.7341 

. 7795 

.5977 

.0379 

. 3652 

. 6347 

28 

52 

12 

33 

. 680 S 

. 3192 

.7302 

. 7826 

.5937 

.0380 

. 3660 

. 6340 

27 

48 

16 

34 

. 6836 

. 3164 

.7263 

. 7858 

.5896 

.0381 

. 3668 

. 6332 

26 

44 

20 

35 

.26864 

.73136 

3.7224 

.27889 

3.5856 

1.0382 

.03676 

.96324 

25 

40 

24 

36 

. 6892 

. 3108 

.7186 

. 7920 

.5816 

.0382 

. 3684 

. 6316 

24 

36 

28 

37 

. 6920 

. 3080 

.7147 

. 7952 

.5776 

.0383 

. 3691 

. 6308 

23 

32 

32 

38 

. 6948 

. 3052 

.7108 

. 7983 

.5736 

.0384 

. 3699 

. 6301 

22 

28 

36 

39 

. 6976 

. 3024 

.7070 

. 8014 

.5696 

.0385 

. 3707 

. 6293 

21 

24 

40 

40 

.27004 

.72996 

3.7031 

.28046 

3.5656 

1.0386 

.03715 

.96285 

20 

20 

44 

41 

. 7032 

. 296S 

.6993 

. 8077 

.5016 

.0387 

. 3723 

. 6277 

19 

16 

48 

42 

. 7060 

. 2940 

.6955 

. 8109 

.5576 

.0387 

. 3731 

. 6269 

18 

12 

52 

43 

. 7088 

. 2912 

.0917 

. 8140 

.5536 

.0388 

. 3739 

. 6261 

17 

8 

56 

44 

. 7110 

. 2884 

.6878 

. 8171 

.5497 

.0389 

. 3746 

. 6253 

16 

4 

3 

45 

.27144 

.72856 

3.0840 

.28203 

3.5457 

1.0390 

.03754 

.96245 

15 

57 

4 

46 

. 7172 

. 2828 

.6802 

. 8234 

.5418 

.0391 

. 3762 

. 6238 

14 

56 

8 

47 

. 7200 

. 2800 

.6765 

. 8266 

.5378 

.0392 

. 3770 

. 6230 

13 

52 

12 

48 

. 7228 

. 2772 

.6727 

. 8297 

.5339 

.0393 

. 3778 

. 6222 

12 

18 

16 

49 

. 7256 

. 2744 

.6689 

. 8328 

.5300 

.0393 

. 3786 

. 6214 

11 

44 

20 

50 

.27284 

.72710 

3.6651 

.28360 

3.5261 

1.0394 

.03794 

.96206 

10 

40 

24 

51 

. 7312 

. 2688 

.0614 

. 8391 

.5222 

.0395 

. 3S02 

. 6198 

9 

36 

28 

52 

. 7340 

. 2660 

.6576 

. 8423 

.5183 

.0396 

. 3S10 

. 6190 

8 

32 

32 

53 

. 7368 

. 2632 

.0539 

. 8454 

.5144 

.0397 

. 3818 

. 6182 

7 

28 

36 

54 

. 7390 

. 2604 

.6502 

. 8486 

.5105 

.0398 

. 3826 

. 6174 

6 

24 

40 

55 

.27424 

.72576 

3.0464 

.2S517 

3.5066 

1.0399 

.03834 

.96166 

5 

20 

44 

56 

. 7452 

. 2548 

.6427 

. 8549 

.5028 

.0399 

. 3842 

. 6158 

4 

16 

48 

57 

. 7480 

. 2520 

.6390 

. 8580 

.4989 

.0400 

. 3850 

. 6150 

3 

12 

52 

58 

. 7508 

. 2492 

.6353 

. 8611 

.4951 

.0401 

. 3858 

. 6142 

2 

8 

56 

59 

. 7530 

. 2464 

.0316 

. 8643 

.4912 

.0402 

. 3866 

. 6134 

1 

4 

4 

60 

. 7504 

. 2436 

.0279 

. 8674 

.4874 

.0403 

. 3874 

. 6126 

0 

56 

M.S. 

M 

Cosine. 

Vrs. Sin. 

Secante. 

Cotang. 

Taugent, 

Cosee’ut 

Vrs.Cos 

Sine. 

M 

M.S. 

7“ 

o 

o< 

o 



N atural. 




74° 

4 b 
































Natural Lines. 


225 


l h 

16° 

Natural Trig 

onometrical 

Functions. 

163° 

10 h 

M.S. 

M 

Sine. IVrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

4 

0 

.27564 

.72436 

3.6279 

.28674 

3.4874 

1.0403 

.03874 

.96126 

60 

56 

4 

1 

. 7592 

. 2408 

.6243 

. 8706 

.4836 

.0404 

. 3882 

. 6118 

59 

56 

8 

2 

. 7620 

. 23S0 

.6206 

. 8737 

.4798 

.0405 

. 3890 

. 6110 

58 

52 

12 

3 

. 7648 

. 2352 

.6169 

. 8769 

.4760 

.0406 

. 3898 

. 6102 

57 

48 

16 

4 

. 7675 

. 2321 

.6133 

. 8800 

.4722 

.0406 

. 3906 

. 6094 

56 

44 

20 

5 

.27703 

.72296 

3.6096 

.28832 

3.4684 

1.0407 

.03914 

.96086 

55 

40 

24 

6 

. 7731 

. 2268 

.6060 

. 8863 

.4646 

.0408 

. 3922 

. 6078 

54 

36 

28 

7 

. 7759 

. 2240 

.6024 

. 8895 

.4608 

.0409 

. 3930 

. 6070 

53 

32 

2.2 

8 

. 7787 

. 2213 

.5987 

. 8926 

.4570 

.0410 

. 3938 

. 6062 

52 

28 

36 

9 

. 7815 

. 2185 

.5951 

. 8958 

.4533 

.0411 

. 3946 

. 6054 

51 

24 

40 

10 

.27843 

.72157 

3.5915 

.28990 

3.4495 

1.0412 

.03954 

.96045 

50 

20 

44 

11 

. 7871 

. 2129 

.5879 

. 9021 

.445S 

.0413 

. 3962 

. 6037 

49 

16 

48 

12 

. 7899 

. 2101 

.5843 

. 9053 

.4420 

.0413 

. 3971 

. 6029 

48 

12 

52 

13 

. 7927 

. 2073 

.5807 

. 90S4 

.4383 

.0414 

. 3979 

. 6021 

47 

8 

56 

14 

. 7955 

. 2045 

.5772 

. 9116 

.4316 

.0415 

. 3987 

. 6013 

46 

4 

5 

15 

.27983 

.72017 

3.5736 

.29147 

3.4308 

1.0116 

.03995 

.96005 

45 

55 

4 

16 

. 8011 

. 1989 

.5700 

. 9179 

.4271 

.0417 

. 4003 

. 5997 

44 

56 

S 

17 

. 8039 

. 1961 

.5665 

. 9210 

.4234 

.0418 

. 4011 

. 5989 

43 

52 

12 

18 

. 8067 

. 1933 

.5629 

. 9242 

.4197 

.0419 

. 4019 

. 5980 

42 

48 

16 

19 

. 8094 

. 1905 

.5594 

. 9274 

.4160 

.0420 

. 4028 

. 5972 

41 

44 

20 

20 

.28122 

.71877 

3.5559 

.29305 

3.4124 

1.0420 

.04036 

.95964 

40 

40 

24 

21 

. 8150 

. 1S49 

.5523 

. 9337 

.4087 

.0421 

. 4044 

. 5956 

39 

36 

28 

22 

. 8178 

. 1822 

.5488 

. 9368 

.4050 

.0422 

. 4052 

. 5948 

38 

32 

32 

23 

. 8206 

. 1794 

.5453 

. 9400 

.4014 

.0423 

. 4060 

. 5940 

37 

28 

36 

24 

. 8234 

. 1766 

.5418 

. 9432 

.3977 

.0424 

. 4069 

. 5931 

36 

24 

40 

25 

.28262 

.71738 

3.5383 

.29463 

3.3941 

1.0425 

.04077 

.95923 

35 

20 

44 

26 

. 8290 

. 1710 

.5348 

. 9495 

.3904 

.0426 

. 4085 

. 5915 

34 

16 

48 

27 

. 8318 

. 1682 

.5313 

. 9526 

.3868 

.0427 

. 4093 

. 5907 

33 

12 

52 

28 

. 8346 

. 1654 

.5279 

. 9558 

.3832 

.0428 

. 4101 

. 5898 

32 

8 

56 

29 

. 8374 

. 1626 

.5244 

. 9590 

.3795 

.0428 

. 4110 

. 5890 

31 

4 

6 

30 

.28401 

.71608 

3.5209 

.29621 

3.3759 

1.0429 

.04118 

.95882 

30 

54 

4 

31 

. 8429 

. 1570 

.5175 

. 9653 

.3723 

.0430 

. 4126 

. 5874 

29 

56 

8 

32 

. 8457 

. 1543 

.5140 

. 9685 

.3687 

.0431 

. 4134 

. 5865 

28 

52 

12 

33 

. 8485 

. 1515 

.5106 

. 9716 

.3651 

.0432 

. 4143 

. 5857 

27 

48 

16 

34 

. 8513 

. 1487 

.5072 

. 9748 

.3616 

.0133 

. 4151 

. 6849 

26 

44 

20 

35 

.28541 

.71459 

3-5037 

.29780 

3.3580 

1.0434 

.04159 

.95840 

25 

40 

24 

36 

. 8569 

. 1431 

.5003 

. 9811 

.3541 

.0435 

. 4168 

. 5832 

24 

36 

28 

37 

. 8597 

. 1403 

.4969 

. 9843 

.3509 

.0436 

. 4176 

. 5824 

23 

32 

32 

38 

. 8624 

. 1375 

.4935 

. 9875 

.3473 

.0437 

. 4184 

. 5816 

22 

28 

36 

39 

. 8652 

. 1347 

.4901 

. 9906 

.3438 

.0438 

. 4193 

. 5807 

21 

24 

40 

40 

.28680 

.71320 

3.4807 

.29938 

3.3402 

1.0438 

.04201 

.95799 

20 

20 

44 

41 

. 8708 

. 1292 

.4833 

. 9970 

.3367 

.0439 

. 4209 

. 5791 

19 

16 

48 

42 

. 8736 

. 1264 

.4799 

.30001 

.3332 

.0440 

. 4218 

. 5782 

18 

12 

52 

43 

. 8761 

. 1236 

.4766 

. 0033 

.3296 

.0441 

. 4226 

. 5774 

17 

8 

56 

44 

. 8792 

. 1208 

.4732 

. 0065 

.3261 

.0442 

. 4234 

. 5765 

16 

4 

7 

45 

.28820 

.71180 

3.4698 

.30096 

3.3226 

1.0443 

. .04243 

.95757 

15 

53 

4 

46 

. 8847 

. 1152 

.4665 

. 0128 

.3191 

.0444 

. 4251 

. 5749 

14 

£6 

8 

47 

. 8875 

. 1125 

.4632 

. 0160 

.3156 

.0445 

. 4260 

. 5740 

13 

52 

12 

48 

. 8903 

. 1097 

.4598 

. 0192 

.3121 

.0446 

. 4268 

. 5732 

12 

48 

16 

49 

. S931 

. 1069 

.4565 

. 0223 

.3087 

.0447 

. 4276 

. 5723 

11 

14 

20 

50 

.28959 

.71011 

3.4532 

.30255 

3.3052 

1.0448 

.04285 

.95715 

10 

40 

24 

51 

. 8987 

. 1013 

.4498 

. 0287 

.3017 

.0448 

. 4293 

. 5707 

9 

36 

28 

52 

. 9014 

. 0985 

.4465 

. 0319 

.2983 

.0449 

. 4302 

. 5698 

8 

32 

32 

53 

. 9042 

. 0958 

.4432 

. 0350 

.2948 

.0450 

. 4310 

. 5690 

7 

28 

36 

54 

. 9070 

. 0930 

.4399 

. 0382 

.2914 

.0451 

. 4319 

. 5681 

6 

24 

40 

55 

.29098 

.70902 

3.4366 

.30414 

3.2879 

1.0452 

.04327 

.95673 

5 

20 

44 

56 

. 9126 

. 0S74 

.4334 

. 0446 

.2845 

.0453 

. 4335 

. 5664 

4 

16 

48 

57 

. 9154 

. 0846 

.4301 

. 0478 

.2811 

.0454 

. 4344 

. 5656 

3 

12 

52 

58 

. 9181 

. 0818 

.4268 

. 0509 

.2777 

.0455 

. 4352 

. 5647 

2 

8 

56 

59 

. 9209 

. 0791 

.4236 

. 0541 

.2742 

.0156 

. 4361 

. 5639 

1 

4 

8 

60 

. 9237 

. 0763 

.4203 

. 0573 

.2708 

.0457 

. 4369 

. 5630 

0 

52 

M.S. 

7 h 

M 

106 

Cosine. 

3 

Vis. Sin. 

Secante. 

Cotang.ITangent. 

Natural. 

Cosec'nt 

Vrs.Cos 

Sine. 

M 

73° 

M.S. 

4 h 


15 






















226 Natural Lines. 


l h 

17° 

Natural Trigonometrical Functions 

162° 

10 h 

M. S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Co tang. 

Secante. 

Vi s. Sin 

Cosine. 

M 

M.S. 

8 

0 

.29237 

.70763 

3.4203 

.30573 

3.2708 

1.0457 

.04369 

.95630 

60 

53 

4 

1 

. 9265 

. 0735 

.4170 

. 0605 

.2674 

.0458 

. 4378 

. 5622 

59 

56 

8 

2 

. 9293 

. 0707 

.4138 

. 0637 

.2640 

.0459 

. 4386 

. 5613 

58 

52 

12 

3 

. 932 L 

. 0679 

.4106 

. 0668 

.2607 

.0460 

. 4395 

. 5605 

57 

48 

16 

4 

. 934S 

. 0651 

.4073 

. 0700 

.2573 

.0461 

. 4404 

. 5596 

56 

44 

20 

5 

.29376 

.70624 

3.4041 

.30732 

3.2539 

1.0461 

.04412 

.95588 

55 

40 

24 

6 

. 9404 

. 0596 

.4009 

. 0764 

.2505 

.0462 

. 4421 

. 5579 

54 

36 

28 

7 

. 9432 

. 0568 

.3977 

. 0796 

.2472 

.0463 

. 4426 

. 5571 

53 

32 

32 

8 

. 9460 

. 0540 

.3945 

. 0828 

.2438 

.0464 

. 4438 

. 5562 

52 

28 

36 

9 

. 9487 

. 0512 

.3913 

. 0859 

.2405 

.0465 

. 4446 

. 5554 

51 

24 

40 

10 

.29515 

.70485 

3.3881 

.30891 

3.2371 

1.0466 

.04455 

.95545 

50 

20 

44 

11 

. 9543 

. 0457 

.3849 

. 0923 

.2338 

.0467 

. 4463 

. 5536 

49 

16 

48 

12 

. 9571 

. 0429 

.3817 

. 0955 

.2305 

.0468 

. 4472 

. 5528 

48 

12 

52 

13 

. 9598 

. 0401 

.3785 

. 0987 

.2271 

.0469 

. 4481 

. 5519 

47 

8 

f>6 

14 

. 9626 

. 0374 

.3751 

. 1019 

.2238 

.0470 

. 4489 

. 5511 

46 

4 

9 

15 

.29654 

770346 

3.3722 

.31051 

3.2205 

1.0471 

.04498 

.95502 

45 

51 

4 

10 

. 9682 

. 6318 

.3690 

. 1083 

.2172 

.0472- 

. 4507 

. 5493 

44 

56 

8 

17 

. 9710 

. 0290 

.3659 

. 1115 

.2139 

.0473 

. 4515 

. 5485 

43 

52 

12 

IS 

. 9737 

. 0262 

.3627 

. 1146 

.2106 

.0474 

. 4524 

. 5476 

42 

48 

16 

19 

. 9765 

. 0235 

.3596 

. 1178 

.2073 

.0475 

. 4532 

. 5467 

41 

44 

20 

20 

.29793 

.70207 

3.3565 

.31210 

3.2041 

1.0476 

.04541 

.95459 

40 

40 

24 

21 

. 9821 

. 0179 

.3534 

. 1242 

.2008 

.0477 

. 4550 

. 5450 

39 

36 

28 

22 

. 9848 

. 0151 

.3502 

. 1274 

.1975 

.0478 

. 4558 

. 5441 

38 

32 

32 

23 

. 9876 

. 0124 

.3471 

. 1306 

.1942 

.0478 

. 4567 

. 5433 

37 

28 

36 

24 

. 9904 

. 0096 

.3440 

. 1338 

.1910 

.0479 

. 4576 

. 5424 

36 

24 

40 

25 

.29932 

.70068 

3.3409 

.31370 

3.1877 

1.0480 

.04585 

.95415 

35 

20 

44 

26 

. 9959 

. 0040 

.3378 

. 1402 

.1845 

.0481 

. 4593 

. 5407 

34 

16 

48 

27 

. 9987 

. 0013 

.3347 

. 1434 

.1813 

.0482 

. 4602 

. 5398 

33 

12 

52 

28 

.30015 

.69982 

.3316 

. 1466 

.1780 

.0483 

. 4611 

. 5389 

32 

8 

56 

29 

.30043 

. 9957 

.3286 

. 1498 

.1748 

.0484 

. 4619 

. 5380 

31 

4 

10 

30 

.30070 

.69929 

3.3255 

.31530 

3.1716 

1.0485 

.04628 

.95372 

30 

50 

4 

31 

. 0098 

. 9902 

.3224 

. 1562 

.1684 

.0486 

. 4637 

. 5363 

29 

56 

8 

32 

. 0126 

. 9874 

.3194 

. 1594 

.1652 

.0487 

. 4646 

. 5354 

28 

52 

12 

33 

. 0154 

. 9846 

.3163 

. 1626 

.1620 

.0488 

. 4654 

. 5345 

27 

48 

16 

34 

. 0181 

. 9818 

.3133 

. 1658 

.158S 

.0489 

. 4663 

. 5337 

-6 

44 

20 

35 

.30209 

.69791 

3.3102 

.31690 

3.1556 

1.0490 

.04672 

.95328 

25 

40 

24 

36 

. 0237 

. 9763 

.3072 

. 1722 

.1524 

.0491 

. 4681 

. 5319 

24 

36 

28 

37 

. 0265 

. 9735 

.3042 

. 1754 

.1492 

.0492 

. 4690 

. 5310 

23 

32 

32 

38 

. 0292 

. 9707 

.3011 

. 1786 

.1460 

.0493 

. 4698 

. 5301 

22 

28 

36 

39 

. 0320 

. 9680 

.2981 

. 1818 

.1429 

.0494 

. 4707 

. 5293 

21 

24 

40 

40 

.30348 

.69652 

3.2951 

.31850 

3.1397 

1.0495 

.04716 

.95284 

20 

20 

44 

41 

. 0375 

. 9624 

.2921 

. 1882 

.1366 

.0496 

. 4725 

. 5275 

19 

16 

48 

42 

. 0403 

. 9597 

.2891 

. 1914 

.1334 

.0497 

. 4734 

. 5266 

18 

12 

52 

43 

. 0431 

. 9569 

.2861 

. 1946 

.1303 

.0498 

. 4743 

. 5257 

17 

8 

56 

44 

. 0459 

. 9541 

.2831 

. 1978 

.1271 

.0499 

. 4751 

. 524S 

16 

4 

11 

45 

.30486 

.69513 

3.2801 

.32010 

3.1210 

1.0500 

.04760 

.95239 

15 

49 

4 

46 

. 0514 

. 9486 

.2772 

. 2042 

.1209 

.0501 

. 4769 

. 5231 

14 

56 

8 

47 

. 0542 

. 9458 

.2742 

. 2074 

.1177 

.0502 

. 4778 

. 5222 

13 

52 

12 

48 

. 0569 

. 9430 

.2712 

. 2106 

1140 

.0503 

. 4787 

. 5213 

12 

48 

16 

49 

. 0597 

. 9403 

.2683 

2138 

.1115 

.0504 

. 4796 

. 5204 

11 

44 

20 

50 

.30625 

.69375 

3.2653 

.32171 

3.1084 

1.0505 

.04805 

.95195 

10 

40 

24 

51 

. 0653 

. 9347 

.2624 

. 2203 

.1053 

.0506 

. 4814 

. 5186 

9 

36 

28 

52 

. 0680 

. 9320 

.2594 

. 2235 

.1022 

.0507 

. 4823 

. 5177 

8 

32 

32 

53 

. 0708 

. 9292 

.2565 

. 2267 

.0991 

.0508 

. 4832 

. 5168 

7 

28 

36 

54 

. 0736 

. 9264 

.2535 

. 2299 

.0960 

.0509 

. 4840 

. 5159 

6 

24 

40 

55 

.30763 

.69237 

3.2506 

.32331 

3.0930 

1.0510 

.04849 

.95150 

5 

20 

44 

56 

. 0791 

. 9209 

.2477 

. 2363 

.0899 

.0511 

. 4858 

. 5141 

4 

16 

48 

57 

. 0819 

. 9181 

.2448 

. 2395 

.0868 

.0512 

. 4867 

. 5132 

3 

12 

52 

58 

. 0846 

. 9154 

.2419 

. 2428 

.0838 

.0513 

. 4876 

. 5124 

2 

8 

56 

59 

. 0874 

. 9126 

.2390 

. 2460 

.0807 

.0514 

. 4885 

. 5115 

1 

4 

13 

60 

. 0902 

. 9098 

.2361 

. 2492 

.0777 

.0515 

. 4894 

. 51U6 

0 

48 

M.S. 

M 

Cosine. 

Yrs. SinJ 

Secante. 

Cotang.[ 

Tangent. 

Cosec’ut 1 

Vrs.Cos 

Sine. 

M 

M.S. 

?h 

107° 



Natural. 




72° 

4 h 

























Natural Lines 


22" 


Natural Trigonometrical 

Functions. 

161° 

10 u 

Sine. I 

Vrs. Cos.! 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

.30902 

.69098 

3.2361 

.32492 

3.0777 

1.0515 

.04894 

.95106 

60 

48 

. 0929 

. 9071 

.2332 

. 2524 

.0746 

.0516 

. 4903 

. 5097 

59 

56 

. 0957 

. 9043 

.2303 

. 2556 

.0716 

.0517 

. 4912 

. 5088 

58 

52 

. 0985 

. 9015 

.2274 

. 2588 

.0686 

.0518 

. 4921 

. 5079 

57 

48 

. 1012 

. 8988 

.2245 

. 2621 

.0655 

.0519 

. 4930 

. 5070 

56 

44 

.31040 

.08960 

3.2216 

.32653 

3.0625 

1.0520 

.04939 

.95061 

55 

40 

. 1008 

. 8932 

.2188 

. 2685 

.0595 

.0521 

. 4948 

. 5051 

54 

36 

. 1095 

. 8905 

.2159 

. 2717 

.0565 

.0522 

. 4957 

. 5042 

53 

32 

. 1123 

. 8877 

.2131 

. 2749 

.0535 

.0523 

. 4966 

. 5033 

52 

28 

. 1150 

. 8849 

.2102 

. 2782 

.0505 

.0524 

. 4975 

. 5024 

51 

24 

.31178 

.68822 

3.2074 

.32814 

3.0475 

1.0525 

.04985 

.95015 

50 

20 

. 1206 

. 8794 

.2045 

. 2846 

.0445 

.0536 

. 4994 

. 5006 

49 

16 

. 1233 

. 8766 

.2017 

. 2878 

.0415 

.0527 

. 5003 

. 4997 

48 

12 

. 1201 

. 8739 

.1989 

. 2910 

.0385 

.0528 

. 5012 

. 4988 

47 

8 

. 1289 

. 8711 

.1960 

. 2943 

.0356 

.0529 

. 5021 

. 4979 

46 

4 

.31310 

.68684 

3.1932 

.32975 

3.0326 

1.0530 

.05030 

.94970 

45 

47 

. 1344 

. 8656 

.1901 

. 3007 

.0296 

.0531 

. 5039 

. 4961 

44 

56 

. 1372 

. 8628 

.1876 

. 3039 

.0367 

.0532 

. 5048 

. 4952 

43 

52 

. 1399 

. 8601 

.1848 

. 3072 

.0237 

.0533 

. 5057 

. 4942 

42 

48 

. 1427 

. 8573 

.1820 

. 3104 

.0208 

.0534 

. 5066 

. 4933. 

41 

44 

.31454 

.68545 

3.1792 

.33136 

3.0178 

1.0535 

.0507 6 

.94924 

40 

40 

. 1482 

. 8518 

.1764 

. 3169 

.0149 

.0536 

. 5085 

. 4915 

39 

36 

. 1510 

. 8490 

.1736 

. 3201 

.0120 

.0537 

. 5094 

. 4906 

38 

32 

. 1537 

. 8463 

.1708 

. 3233 

.0090 

.0538 

. 5103 

. 4897 

37 

28 

. 1565 

. 8435 

.1681 

. 3265 

.0061 

.0539 

. 5112 

. 4888 

36 

24 

.31592 

.68407 

3.1653 

.33298 

3.0032 

1.0540 

.05121 

.94878 

35 

20 

. 1020 

. 8380 

.1625 

. 3330 

3.0003 

.0541 

. 5131 

. 4869 

34 

16 

. 1048 

. 8352 

.1598 

. 3362 

2.9974 

.0542 

. 5140 

. 4860 

33 

12 

. 1075 

. 8325 

.1570 

. 3395 

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.0543 

. 5149 

. 4851 

32 

8 

. 1703 

. 8297 

.1543 

. 3427 

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. 5158 

. 4841 

31 

4 

.31730 

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3.1515 

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2.9887 

1.0545 

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30 

46 

. 1758 

. 8242 

.1488 

. 3492 

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. 5177 

. 4823 

29 

56 

. 1786 

. 8214 

.1461 

. 3524 

.9829 

.0547 

. 5186 

. 4814 

28 

52 

. 1813 

. 8187 

.1433 

. 3557 

.9800 

.0548 

. 5195 

. 4805 

27 

48 

. 1841 

. 8159 

.1406 

. 3589 

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.0549 

. 5205 

. 4795 

26 

44 

.31868 

.68132 

3.1379 

.33621 

2.9743 

1.0550 

.05214 

.94786 

25 

40 

. 1890 

. 8104 

.1352 

. 3654 

.9714 

.0551 

. 5223 

. 4777 

24 

36 

. 1923 

. 8076 

.1325 

. 3686 

.9686 

.0552 

. 5232 

. 4767 

23 

32 

. 1951 

. 8049 

.1298 

. 3718 

.9657 

.0553 

. 5242 

. 4758 

22 

28 

. 1978 

. 8021 

.1271 

. 3751 

.9629 

.0554 

. 5251 

. 4749 

21 

24 

.32006 

.67994 

3.1244 

.33783 

2.9600 

1.0555 

.05260 

.94740 

20 

20 

. 2034 

. 7966 

.1217 

. 3816 

.9572 

.0556 

. 5270 

. 4730 

19 

16 

. 2061 

. 7939 

.1190 

. 3848 

.9544 

.0557 

. 5279 

. 4721 

18 

12 

. 2089 

. 7911 

.1163 

. 3880 

.9515 

.0558 

. 5288 

. 4712 

17 

8 

. 2116 

. 7884 

.1137 

. 3913 

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.0559 

. 5297 

. 47(>2 

16 

4 

.32144 

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3.1110 

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2.9459 

1.0560 

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15 

45 

. 2171 

. 7828 

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14 

56 

. 2199 

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13 

52 

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9375 

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12 

48 

. 2254 

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4075 

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11 

44 

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3.0977 

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2.9319 

1.0566 

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10 

40 

. 2309 

. 7691 

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9 

36 

. 2337 

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8 

32 

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7 

28 

. 2392 

. 7608 

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6 

24 

.32419 

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3.0846 

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2.9180 

1.0571 

.05401 

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5 

20 

. 2447 

. 7553 

.0820 

. 4303 

.9152 

.0572 

. 5410 

. 4590 

4 

16 

. 2474 

. 7526 

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. 4335 

.9125 

.0573 

. 5420 

. 4580 

3 

12 

. 2502 

. 7498 

.0767 

. 4368 

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.0574 

. 5429 

. 4571 

2 

8 

. 2529 

. 7471 

.0741 

. 4400 

.9069 

.0575 

. 5439 

. 4561 

1 

4 

. 2557 

. 7443 

.0715 

. 4433 

.9042 

.0576 

. 5448 

. 4552 

o 

44 

Cosine. 

Vrs.Siu.i Secaute. 

Cotung. 

Nati 

Tangent, 

iral. 

Cosec'ut 

Vrs. Cos 

Sine. 

M 

71° 

M.S. 

4 h 





























223 


Natural Lines. 


f 

l h 

19° 

Natural Trig 

onometrical 

Functions. 

160° 

10 h 

M.S. 

M 

Sine. 

Vrs.Oos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

jVrs. Sin 

Cosine. 

M 

M.S. 

10 

0 

.32557 

.67443 

3.0715 

.34433 

2.9042 

1.0576 

! .05448 

.94552 

60 

44 

4 

1 

. 2584 

. 7416 

.0690 

. 4465 

.9015 

.0577 

. 545S 

. 4542 

59 

56 

8 

2 

. 2612 

. 7388 

.0664 

. 4498 

.8987 

.0578 

. 5467 

. 4533 

58 

52 

12 

3 

. 2639 

. 7361 

.0638 

. 4530 

.8900 

.0579 

. 5476 

. 4523 

57 

48 

16 

4 

. 2667 

. 7333 

.0612 

. 4563 

.8933 

.0580 

. 5486 

. 4514 

56 

44 

20 

5 

.32694 

.67306 

3.0586 

.34595 

2.8905 

1.0581 

.05495 

.94504 

55 

40 

24 

6 

. 2722 

. 7278 

.0561 

. 4628 

.8878 

.0582 

. 5505 

. 4495 

54 

36 

28 

7 

. 2749 

. 7251 

.0535 

. 4601 

.8851 

.0584 

. 5515 

. 4485 

53 

32 

32 

8 

. 2777 

. 7223 

.0509 

. 4693 

.8824 

.0585 

. 5524 

. 4476 

52 

28 

36 

9 

. 2804 

. 7196 

.0484 

. 4726 

.8797 

.0586 

. 5534 

. 4466 

51 

24 

i 40 

10 

.32832 

67168 

3.0458 

.34758 

2.8770 

1.0587 

.05543 

.94457 

50 

20 

44 

11 

. 2859 

. 7141 

.0433 

. 4791 

.8743 

.0588 

. 5558 

. 4447 

49 

16 

48 

12 

. 2887 

. 7113 

.0407 

. 4824 

.8716 

.0589 

. 5562 

. 4438 

48 

12 

52 

13 

. 2914 

. 7086 

.0382 

. 4856 

.8689 

.0590 

. 5572 

. 4428 

47 

8 

56 

14 

. 2942 

. 7058 

.0357 

. 4889 

.8662 

.0591 

. 5581 

. 4418 

46 

4 

17 

15 

.32969 

.67031 

3.0331 

.34921 

2.8636 

1.0592 

.05591 

.94409 

45 

43 

4 

16 

. 2996 

. 7003 

.0306 

. 4954 

.861.9 

.0593 

. 5601 

. 4399 

44 

56 

8 

17 

. 3024 

. 6976 

.0281 

. 4987 

.8582 

.0594 

. 5610 

. 4390 

43 

52 

12 

IS 

. 3051 

. 6918 

.0256 

. 5019 

.8555 

.0595 

. 5620 

. 4380 

42 

48 

16 

19 

. 3079 

. 6921 

.0231 

. 5052 

.8529 

.0596 

. 5629 

. 4370 

41 

44 

20 

20 

.33106 

.66894 

3.0206 

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2.8502 

1.0598 

.05639 

.94361 

40 

40 

24 

21 

. 3134 

. 6866 

.0181 

. 5117 

.8476 

.0599 

. 5649 

. 4351 

39 

36 

28 

22 

. 3161 

. 6839 

.0156 

. 5150 

.8449 

.0600 

. 5658 

. 4341 

38 

32 

32 

23 

. 3189 

. 6811 

.0131 

. 5183 

.8423 

.0601 

. 5668 

. 4332 

37 

28 

36 

24 

. 3216 

. 6784 

.0106 

. 5215 

.8390 

.0602 

. 5678 

. 4322 

36 

24 

40 

25 

.33243 

.66756 

3.0081 

.35248 

2.8370 

1.0603 

.05687 

.94313 

35 

20 

44 

26 

. 3271 

. 6729 

.0056 

. 5281 

.8344 

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. 5697 

. 4303 

34 

16 

48 

27 

. 3298 

. 6701 

.0031 

. 5314 

.8318 

.0605 

. 5707 

4293 

33 

12 

52 

28 

. 3326 

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. 5716 

. 4283 

32 

8 

56 

29 

. 3353 

. 6647 

2.9982 

. 5379 

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. 5726 

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31 

4 

18 

30 

.33381 

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2.9957 

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2.8239 

1.0608 

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.94264 

30 

42 

4 

31 

. 3408 

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29 

56 

8 

32 

. 3435 

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. 5477 

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. 4245 

28 

52 

12 

33 

. 3463 

. 6537 

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. 5510 

.8161 

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. 5765 

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27 

48 

16 

34 

. 3490 

. 6510 

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. 5775 

. 4225 

26 

44 

20 

35 

.33518 

.66482 

2.9835 

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2.8109 

1.0614 

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.94215 

25 

40 

24 

36 

. 3545 

. 6455 

.9810 

. 5608 

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. 5794 

. 4206 

24 

36 

28 

37 

. 3572 

. 6427 

.9786 

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.8057 

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. 5804 

. 4196 

23 

32 

32 

38 

. 3600 

. 6400 

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. 5814 

. 4186 

22 

28 

36 

39 

. 3627 

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.6618 

. 5823 

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21 

21 

40 

40 

.33655 

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2.9713 

.35739 

2.7980 

1.0619 

.05833 

.94167 

20 

20 

44 

41 

. 3682 

. 6318 

.9689 

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. 5843 

. 4157 

19 

16 

48 

42 

. 3709 

. 6290 

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. 5805 

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. 5853 

. 4147 

18 

12 

52 

43 

. 3737 

. 6263 

.9641 

. 5838 

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. 5863 

. 4137 

17 

8 

56 

44 

. 3764 

. 6236 

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. 5871 

.7878 

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. 5872 

. 4127 

16 

4 

19 

45 

.33792 

.66208 

2.9593 

.35904 

2.7852 

1.0625 

.05882 

.94118 

15 

41 

4 

46 

. 3819 

. 6181 

.9509 

. 5936 

.7827 

.0626 

. 5892 

. 4108 

14 

56 

8 

47 

. 3846 

. 6153 

.9545 

. 5969 

.7801 

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. 5902 

. 4098 

13 

52 

12 

48 

. 3874 

. 6126 

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. 6002 

.7776 

.0628 

. 5912 

. 4088 

12 

48 

16 

49 

. 3901 

. 6099 

.9497 

. 6035 

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.0629 

. 5922 

. 4078 

11 

44 

20 

50 

.33928 

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2.9474 

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2.7725 

1.0630 

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10 

40 

24 

51 

. 3956 

. 6044 

.9450 

. 6101 

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. 5941 

. 4058 

9 

36 

28 

52 

. 3983 

. 6017 

.9426 

. 6134 

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. 5951 

. 4049 

8 

32 

32 

53 

. 4011 

. 5989 

.9402 

. 6167 

.7650 

.0634 

. 5961 

. 4039 

7 

28 

36 

54 

. 4038 

. 5962 

.9379 

. 6199 

.7625 

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. 5971 

. 4029 

6 

24 

40 

55 

.34065 

.65935 

2.9355 

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2.7600 

1.0636 

.05981 

.94019 

5 

20 

44 

56 

. 4093 

. 5907 

.9332 

. 6265 

.7574 

.0637 

. 6991 

. 4009 

4 

16 

48 

57 

. 4120 

. 5880 

.9308 

. 6298 

.7549 

.0638 

. 6001 

. 3999 

3 

12 

52 

58 

. 4147 

. 5853 

.9285 

. 6331 

.7524 

.0639 

. 6011 

. 3989 

2 

8 

56 

59 

. 4175 

. 6825 

.9261 

. 6364 

.7500 

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. 6021 

. 3979 

1 

4 

20 

60 

. 4202 

. 5798 

.9238 

. 6397 

.7475 

.0642 

. 6031 

. 3969 

0 

40 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secante, 

Cotang. 

Tangent. 

Cosec’nt 

Vrs.Cos 

Sine. 

M 

M.S. 

7 h 

109 

o 



Natural. 




70° 

4 h 























Natural Lines. 229 


l h 

20 c 

Natural Trig 

onometrical Fmictions 

159° 

10 h 

M.S 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

30 

0 

.34202 

.65798 

2.9238 

.36397 

2.7475 

1.0642 

.06031 

.93909 

60 

40 

4 

1 

. 4229 

. 5771 

.9215 

. 6430 

.7450 

.0643 

. 6041 

. 3959 

59 

56 

8 

2 

. 4257 

. 5743 

.9191 

. 6463 

.7425 

.0644 

. 6051 

. 3949 

58 

52 

12 

3 

. 42S4 

. 5716 

.9168 

. 6496 

.7400 

.0645 

. 6061 

. 3939 

57 

48 

16 

4 

. 4311 

. 5689 

.9145 

. 6529 

.7376 

.0646 

. 6071 

. 3929 

56 

44 

20 

5 

.34339 

.65661 

2.9122 

.36562 

2.7351 

1.0647 

.06080 

.93919 

55 

40 

24 

6 

. 4366 

. 5634 

.9098 

. 6595 

.7326 

.0648 

. 6090 

. 3909 

54 

36 

28 

7 

. 4393 

. 5607 

.9075 

. 6628 

.7302 

.0650 

. 6100 

. 3899 

53 

32 

32 

8 

. 4421 

. 5579 

.9052 

. 6661 

.7277 

.0651 

. 6110 

. 3889 

52 

28 

36 

9 

. 4448 

. 5552 

.9029 

. 6694 

.7252 

.0652 

. 6121 

. 3879 

51 

24 

40 

10 

.34475 

.65525 

2.9006 

.36727 

2.7228 

1.0653 

.06131 

.93S69 

50 

20 

44 

11 

. 4502 

. 5497 

.8983 

. 6760 

.7204 

.0654 

. 6141 

. 3859 

49 

16 

48 

12 

. 4530 

. 5470 

.8960 

. 6793 

.7179 

.0655 

. 6151 

. 3849 

48 

12 

62 

13 

. 4557 

. 5443 

.8937 

. 6826 

.7155 

.0656 

. 6161 

. 3839 

47 

8 

56 

14 

. 4584 

. 5415 

.8915 

. 6859 

.7130 

.0658 

. 6171 

. 3829 

46 

4 

31 

15 

.34612 

.65388 

2.8S92 

.36892 

2.7106 

1.0659 

.06181 

.93819 

45 

39 

4 

16 

. 4639 

. 5361 

.8869 

. 6925 

.7082 

.0660 

. 6191 

. 3809 

44 

56 

8 

17 

. 4666 

. 5334 

.8846 

. 6958 

.7058 

.0661 

. 6201 

. 3799 

43 

52 

12 

18 

. 4693 

. 5306 

.8824 

. 6991 

.7033 

.0662 

. 6211 

. 3789 

42 

48 

16 

19 

. 4721 

. 5279 

.8801 

. 7024 

.7009 

.0663 

. 6221 

. 3779 

41 

44 

20 

20 

.34748 

.65252 

2.8778 

.37057 

2.6985 

1.0664 

.06231 

.93769 

40 

40 

24 

21 

. 4775 

. 5225 

.8756 

. 7090 

.6961 

.0666 

. 6241 

. 3758 

39 

36 

28 

22 

. 4803 

. 5197 

.8733 

. 7123 

.6937 

.0667 

. 6251 

. 3748 

38 

32 

32 

23 

. 4830 

. 5170 

.8711 

. 7156 

.6913 

.0668 

. 6262 

. 3738 

37 

28 

36 

24 

. 4857 

. 5143 

.8688 

. 7190 

.6S89 

.0669 

. 6272 

. 3728 

36 

24 

10 

25 

.34884 

.65115 

2.8666 

.37223 

2.6865 

1.0670 

.06282 

.93718 

35 

20 

44 

26 

. 4912 

. 5088 

.8641 

. 7256 

.6841 

.0671 

. 6292 

. 3708 

34 

16 

48 

27 

. 4939 

. 5061 

.8621 

. 7289 

.6817 

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. 6302 

. 3698 

33 

12 

52 

28 

. 4966 

. 5034 

.8599 

. 7322 

.6794 

.0674 

. 6312 

. 3687 

32 

8 

56 

29 

. 4993 

. 5006 

.8577 

. 7355 

.6770 

.0675 

. 6323 

. 3677 

31 

4 

33 

30 

.35021 

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2.8554 

.37388 

2.6746 

1.0676 

.06333 

.93667 

30 

38 

4 

31 

. 5048 

. 4952 

.8532 

. 7422 

.6722 

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. 6343 

. 3657 

29 

56 

8 

32 

. 5075 

. 4925 

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. 6353 

. 3647 

28 

52 

12 

33 

. 5102 

. 4897 

.8488 

. 7488 

.6675 

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. 6363 

. 3637 

27 

48 

16 

34 

. 5130 

. 4870 

.8466 

. 7521 

.6652 

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. 6373 

. 3626 

26 

44 

20 

35 

.35157 

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2.8444 

.37554 

2.6628 

1.0682 

.06384 

.93616 

25 

40 

24 

36 

. 5184 

. 4816 

.8422 

. 7587 

.6604 

.0683 

. 6394 

. 3606 

24 

36 

28 

37 

. £211 

. 4789 

.8400 

. 7621 

.6581 

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. 6404 

. 3596 

23 

32 

32 

38 

. 5239 

. 4761 

.8378 

. 7654 

.6558 

.0685 

. 6414 

. 3585 

22 

28 

36 

39 

. 5266 

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.6534 

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. 6425 

. 3575 

21 

24 

40 

40 

.35293 

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2.S334 

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2.6511 

1.0688 

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.93565 

20 

20 

44 

41 

. 5320 

. 4680 

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. 6445 

. 3555 

19 

16 

48 

42 

. 5347 

. 4652 

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. 3544 

18 

12 

52 

43 

. 5375 

. 4625 

.8269 

. 7820 

.6441 

.0691 

. 6466 

. 3534 

17 

8 

56 

44 

. 5402 

. 4598 

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. 7853 

.6418 

.0692 

. 6476 

. 3524 

16 

4 

33 

45 

.35429 

.64571 

2.8225 

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2.6394 

1.0694 

.06486 

.93513 

15 

37 

4 

46 

. 5456 

. 4544 

.8204 

. 7920 

.6371 

.0695 

. 6497 

. 3503 

14 

56 

8 

47 

. 5483 

. 4516 

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. 7953 

.6348 

.0696 

, 6507 

. 3493 

13 

52 

12 

48 

. 5511 

. 4489 

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. 7986 

.6325 

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. 6517 

. 3482 

12 

48 

16 

49 

. 5538 

. 4462 

.8139 

8020 

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. 6528 

. 3472 

11 

44 

20 

50 

.35565 

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2.S117 

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2.6279 

1.0699 

.06538 

.93462 

10 

40 

24 

51 

. 5592 

. 4408 

.8U96 

. 8086 

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.0701 

. 6548 

. 3451 

9 

36 

28 

52 

. 5619 

. 4380 

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. S120 

.6233 

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. 6559 

. 3441 

8 

32 

32 

53 

. 5647 

. 4353 

.8053 

. 8153 

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. 6569 

. 3431 

7 

28 

36 

54 

. 5674 

. 4326 

.8032 

. 8186 

.6187 

.0704 

. 6579 

. 3420 

6 

24 

40 

55 

.35701 

.64299 

2.8010 

.38220 

2.6164 

1.0705 

.06590 

.93410 

5 

20 

44 

56 

. 5728 

. 4272 

.7989 

. 8253 

.6142 

.0707 

. 6600 

. 3400 

4 

16 

48 

57 

. 5755 

. 4245 

.7968 

. 8286 

.6119 

.0708 

. 6611 

. 3389 

3 

12 

52 

58 

. 5782 

. 4217 

.7947 

. 8320 

.6096 

.0709 

. 6621 

. 3379 

2 

8 

5(5 

59 

. 5810 

. 4190 

.7925 

. 8353 

.6073 

.0710 

. 6631 

. 3368 

1 

4 

31 

60 

. 5837 

. 4163 

.7904 

. 8386 

.6051 

.0711 

. 6642 

. 3358 j 

0 

30 

M.S. 

M 

Cosine. 

Vrs.Sin.l Secaute. 

Cotang. [Tangent, 

Cosec'nt 1 Yrs.Cos 

Sine. | 

M 

M.S. 

7 h 

110° 



Natural. 




69° 

4 h 

























230 


Natural Lines. 


l h 

21 c 

Natural Trigonometrical Functions. 

158° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

SCosec’nte 

Tang. 

Cotang. 

Secante 

Vrs.Sin 

Cosine. 

j M 

M.S. 

34 

0 

.35837 

.64163 

| 2.7904 

.38386 

2.6051 

1.0711 

.06642 

.93358 

60 

36 

4 

l 

. 6864 

. 4136 

.7883 

. 8420 

.6028 

.0713 

. 6652 

. 3348 

59 

56 

8 

2 

. 5891 

. 4109 

.7862 

. 8453 

.6006 

.0714 

. 6663 

. 3337 

58 

52 

12 

3 

. 5918 

. 4082 

.7841 

. 8486 

.5983 

.0715 

. 6673 

. 3327 

57 

48 

16 

4 

. 5945 

. 4055 

.782(1 

. 8520 

.5960 

.0716 

. 6684 

. 3316 

56 

44 

20 

5 

.35972 

.64027 

2.7799 

.38553 

2.5938 

1.0717 

.06694 

' .93306 

55 

40 

24 

6 

. 6000 

. 4000 

.7778 

. 8587 

.5916 

.0719 

. 6705 

. 3295 

54 

36 

28 

7 

. 6027 

. 3973 

.7757 

. 8620 

.5893 

.0720 

. 6715 

. 3285 

53 

32 

32 

8 

. 6054 

. 3946 

.7736 

. 8654 

.5871 

.0721 

. 6726 

. 3274 

52 

28 

36 

9 

. 6081 

. 3919 

.7715 

. 8687 

.5848 

.0722 

. 6736 

. 3264 

51 

24 

40 

10 

.36108 

.63892 

2.7694 

.38720 

2.5826 

1.0723 

.06747 

.93253 

50 

20 

44 

11 

. 6135 

. 3865 

.7674 

. 8754 

.5804 

.0725 

. 6757 

. 3243 

49 

16 

48 

12 

. 6162 

. 3837 

.7653 

. 8787 

.5781 

.0726 

. 6768 

. 3232 

48 

12 

62 

13 

. 6189 

. 3810 

.7632 

. 8S21 

.5759 

.0727 

. 6778 

. 3222 

47 

8 

66 

14 

. 6217 

. 3783 

.7611 

. 8854 

.5737 

.0728 

. 6789 

. 3211 

46 

4 

35 

15 

.36244 

.63756 

2.7591 

.38888 

2.5715 

1.0729 

.06799 

.93201 

45 

35 

4 

16 

. 6271 

. 3729 

.7570 

. 8921 

.5693 

.0731 

. 6810 

. 3190 

44 

56 

8 

17 

. 6298 

. 3702 

.7550 

. 8955 

.5671 

.0732 

. 6820 

. 3180 

43 

52 

12 

18 

. 6325 

. 3675 

.7529 

. 8988 

.5649 

.0733 

. 6831 

. 3169 

42 

48 

16 

19 

. 6352 

. 3648 

.7509 

. 9022 

.5627 

.0734 

. 6841 

. 3158' 

41 

44 

20 

20 

.36379 

.63621 

2.7488 

.39055 

2.5605 

1.0736 

.06852 

.93148 

40 

40 

24 

21 

. 6406 

. 3593 

.7468 

. 9089 

.5583 

.0737 

. 6863 

. 3137 

39 

36 

28 

22 

. 6433 

. 3566 

.7447 

. 9122 

.5561 

.0738 

. 6873 

. 3127 

38 

32 

32 

23 

. 6460 

. 3539 

.7427 

. 9156 

.5539 

.0739 

. 6884 

. 3116 

37 

28 

36 

24 

. 6488 

. 3512 

.7406 

. 9189 

.5517 

.0740 

. 6894 

. 3105 

36 

24 

40 

25 

.36515 

.63485 

2.7386 

.39223 

2.5495 

1.0742 

.06905 

.93095 

35 

20 

44 

26 

. 6542 

. 3458 

.7366 

. 9257 

.5473 

.0743 

. 6916 

. 3084 

34 

16 

48 

27 

. 6569 

. 3431 

.7346 

. 9290 

.5451 

.0744 

. 6926 

3074 

33 

12 

52 

28 

. 6596 

. 3404 

.7325 

. 9324 

.5430 

.0745 

. 6937 

. 3063 

32 

8 

56 

29 

. 6623 

. 3377 

.7305 

. 9357 

.5408 

.0747 

. 6947 

. 3052 

31 

4 

36 

30 

.36650 

.63350 

2.7285 

.39391 

2.5386 

1.0748 

.06958 

.93042 

30 

344 

4 

31 

. 6677 

. 3323 

.7265 

. 9425 

.5365 

.0749 

. 6969 

. 3031 

29 

56 

8 

32 

.' 6704 

. 3296 

.7245 

. 9458 

.5343 

.0750 

. 6979 

. 3020 

28 

52 

12 

33 

. 6731 

. 3269 

.7225 

. 9492 

.5322 

.0751 

. 6990 

. 3010 

27 

48 

16 

34 

. 6758 

. 3242 

.7205 

. 9525 

.5300 

.0753 

. 7001 

. 2999 

26 

44 

20 

35 

.36785 

.63214 

2.7185 

.89559 

2.5278 

1.0754 

.07012 

.92988 

25 

40 

24 

36 

. 6812 

. 3187 

.7165 

. 9593 

.5257 

.0755 

. 7022 

. 2978 

24 

36 

28 

37 

. 6839 

. 3160 

.7145 

. 9626 

.5236 

.0756 

. 7033 

. 2967 

23 

32 

32 

38 

. 6866 

. 3133 

.7125 

. 9660 

.5214 

.0758 

. 7044 

. 2956 

22 

28 

36 

39 

. 6893 

. 3106 

.7105 

. 9694 

.5193 

.0759 

. 7054 

. 2945 

21 

24 

40 

40 

.36921 

.63079 

2.7085 

.39727 

2.5171 

1.0760 

.07065 

.92935 

20 

20 

44 

41 

. 6948 

. 3052 

.7065 

. 9761 

.5150 

.0761 

. 7076 

. 2924 

19 

16 

48 

42 

. 6975 

. 3025 

.7045 

. 9795 

.5129 

.0763 

. 7087 

. 2913 

18 

12 

52 

43 

. 7002 

. 2998 

.7026 

. 9828 

.51U8 

.0764 

. 7097 

. 2902 

17 

8 

56 

44 

. 7029 

. 2971 

.7006 

. 9862 

.5086 

.0765 

. 7108 

. 2892 

16 

4 

27 

45 

.37056 

.62944 

2.6986 

.39896 

2.5065 

1.0766 

.07119 

.92881 

15 

33 

4 

46 

. 7083 

. 2917 

.6967 

. 9930 

.5044 

.0768 

. 7130 

. 2870 

14 

56 

8 

47 

. 7110 

. 2890 

.6947 

. 9963 

.5023 

.0769 

. 7141 

. 2859 

13 

52 

12 

48 

. 7137 

. 2863 

.6927 

. 9997 

.5002 

.0770 

. 7151 

. 2848 

12 

48 

16 

49 

. 7164 

. 2836 

.6908 

.40031 

.4981 

.0771 

. 7162 

. 2838 

11 

44 

2C 

50 

.37191 

.62809 

2.6888 

.40065 

2.4960 

1.0773 

.07173 

.92827 

10 

40 

24 

51 

. 7218 

. 2782 

.6869 

. 0098 

.4939 

.0774 

. 7184 

. 2816 

9 

36 

28 

52 

. 7245 

. 2755 

.6849 

. 0132 

.4918 

.0775 

. 7195 

. 2805 

8 

32 

32 

53 

. 7272 

. 2728 

.6830 

. 0166 

.4897 

.0776 

. 7205 

. 2794 

7 

28 

36 

54 

. 7299 

. 2701 

.6810 

. 0200 

.4876 

.0778 

. 7216 

. 2784 

6 

24 

40 

55 

.37326 

.62674 

2.6791 

.40233 

2.4855 

1.0779 

.07227 

.92773 

5 

20 

41 

56 

. 7353 

. 2647 

.6772 

. 0267 

.4834 

.0780 

. 7238 

. 2762 

4 

16 

48 

57 

. 7380 

. 2620 

.6752 

. 0301 

.4813 

.0781 

. 7249 

. 2751 

3 

12 

52 

58 

. 7407 

. 2593 

.6733 

. 0335 

.4792 

.0783 

. 7260 

. 2740 

2 

8 

56 

59 

. 7434 

. 2566 

.6714 

. 0369 

.4772 

.0784 

. 7271 

. 2729 

1 

4 

38 

60 

. 7461 

. 2539 

.6695 

. 0403 

.4751 

.0785 

. 7282 

. 2718 

0 

32 

M. S. 

M 

Cosine. 

Vrs.Sin. 

Seeaute. 

Co tang. 

Tangent. 

Ccsec'ut 

Vrs.Cos 

Sine. 

M i 

M.S. 

7» 

111 0 


• 

Natural. 



68 °| 

4 h 




























Natural Lines. 231 


l h 

22 c 

Natural Trigonometrical Functions 

157° 

10 h 

M. S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante.|Vrs. Sin 

Cosine. 

M 

M.S. 

38 

0 

.37461 

.62539 

2.6695 

.40403 

2.4751 

1.0785 

.07282 

.92718 

60 

33 

4 

1 

. 7488 

. 2512 

.6675 

. 0436 

.4 730 

.0787 

. 7292 

. 2707 

59 

56 

8 

2 

. 7514 

. 2485 

.6656 

. (1170 

.4709 

.0788 

. 7303 

. 2696 

58 

52 

12 

3 

. 7541 

. 2458 

.6637 

. 0504 

.4689 

.0789 

. 7314 

. 2686 

57 

48 

16 

4 

. 7568 

. 2431 

.6618 

. 0538 

.4668 

.0790 

. 7325 

. 2675 

56 

44 

20 

5 

.37595 

.62404 

2.6599 

.40572 

2.4647 

1.0792 

.07336 

.92664 

55 

40 

24 

6 

. 7622 

. 2377 

.6580 

. 0606 

.4627 

.0793 

. 7347 

. 2653 

54 

36 

28 

7 

. 7619 

. 2351 

.6561 

. 0640 

.4606 

.0794 

. 7358 

. 2642 

53 

32 

32 

8 

. 7676 

. 2324 

.6542 

. 0673 

.4586 

.0795 

. 7369 

. 2631 

52 

28 

36 

9 

. 7703 

. 2297 

.6523 

. 0707 

.4565 

.0797 

. 7380 

. 2620 

51 

24 

40 

10 

.37730 

.62270 

2.6504 

.40741 

2.4545 

1.0798 

.07391 

.92609 

50 

20 

44 

11 

. 7757 

. 2243 

.6485 

. 0775 

.4525 

.0799 

. 7402 

. 2598 

49 

16 

48 

12 

. 7784 

. 2216 

.6466 

. 0809 

.4504 

.0801 

. 7413 

. 2587 

48 

12 

52 

13 

. 7811 

. 2189 

.6447 

. 0843 

.44S4 

.0802 

. 7424 

. 2576 

47 

8 

56 

14 

. 7838 

. 2162 

.6428 

. 0877 

.4463 

.0803 

. 7435 

. 2565 

46 

4 

29 

15 

.37865 

.62135 

2.6410 

.40911 

2.4443 

1.0804 

.07446 

.92554 

45 

31 

4 

16 

. 7892 

. 2108 

.6391 

. 0945 

.4423 

.0S06 

. 7457 

. 2543 

44 

56 

8 

17 

. 7919 

. 2081 

.6372 

. 0979 

.4403 

.0807 

. 7468 

. 2532 

43 

52 

12 

18 

. . 7946 

. 2054 

.6353 

. 1013 

.4382 

.0808 

. 7479 

. 2521 

42 

48 

16 

19 

. 7972 

. 2027 

.6335 

. 1047 

.4362 

.0810 

. 7490 

. 2510 

41 

44 

20 

20 

.37999 

.62000 

2.6316 

.41081 

2.4342 

1.0811 

.07501 

.92499' 

40 

4o 

24 

21 

. 8026 

. 1974 

.6297 

. 1115 

.4322 

.0812 

. 7512 

. 2488 

39 

36 

28 

22 

. 8053 

. 1947 

.6279 

. 1149 

.4302 

.0813 

. 7523 

. 2477 

38 

32 

32 

23 

. 8080 

. 1920 

.6260 

. 1183 

.4282 

.0815 

. 7534 

. 2466 

37 

28 

36 

24 

. 8107 

. 1893 

.6242 

. 1217 

.4262 

.0816 

. 7545 

. 2455 

36 

24 

40 

25 

.38134 

.61866 

2.6223 

.41251 

2.4242 

1.0817 

.07556 

.92443 

35 

20 

44 

26 

. 8161 

. 1839 

*6205 

. 1285 

.4222 

.0819 

. 7567 

. 2432 

34 

16 

48 

27 

. 8188 

. 1812 

.6186 

. 1319 

.4202 

.0820 

. 7579 

. 2421 

33 

12 

52 

28 

. 8214 

. 1785 

.6168 

. 1353 

.4182 

.0821 

. 7590 

. 2410 

32 

8 

56 

29 

. 8241 

. 1758 

.6150 

. 1387 

.4162 

.0823 

. 7601 

. 2399 

31 

4 

30 

30 

.38268 

.61732 

2.6131 

.41421 

2.4142 

1.0824 

.07612 

.92388 

30 

30 

4 

31 

. 8295 

. 1705 

.6113 

. 1455 

.4122 

.0825 

. 7623 

. 2377 

29 

56 

8 

32 

. 8322 

. 1678 

.6095 

. 1489 

.4102 

.0826 

. 7634 

. 2366 

28 

52 

12 

33 

. 8349 

. 1651 

.6076 

. 1524 

.4083 

.0828 

. 7645 

. 2354 

27 

48 

16 

34 

. 8376 

. 1624 

.6058 

. 1558 

.4063 

.0829 

. 7657 

. 2343 

26 

44 

20 

35 

.38403 

.61597 

2.6040 

.41592 

2.4043 

1.0830 

.07668 

.92332 

26 

40 

24 

36 

. 8429 

. 1570 

.6022 

. 1626 

.4023 

.0832 

. 7679 

. 2321 

24 

36 

28 

37 

. 8456 

. 1544 

.6003 

. 1660 

.4004 

.0833 

. 7690 

. 2310 

23 

32 

32 

38 

. 8483 

. 1517 

.5985 

. 1694 

.3984 

.0834 

. 7701 

. 2299 

22 

28 

36 

39 

. 8510 

. 1490 

.5967 

. 1728 

.3964 

.0836 

. 7712 

. 2287 

21 

24 

40 

40 

.38537 

.61463 

2.5949 

.41762 

2.3945 

1.0837 

.07724 

.92276 

20 

20 

44 

41 

. 8564 

. 1436 

.5931 

. 1797 

.3925 

.0838 

. 7735 

. 2265 

19 

16 

48 

42 

. 8591 

. 1409 

.5913 

. 1831 

.3906 

.0840 

. 7746 

. 2254 

18 

12 

52 

43 

. 8617 

. 1382 

.5895 

. 1865 

.3886 

.0841 

. 7767 

. 2242 

17 

8 

56 

44 

. 8644 

. 1356 

.5877 

. 1899 

.3867 

.0842 

. 7769 

. 2231 

16 

4 

31 

45 

.38671 

.61329 

2.5859 

.41933 

2.3847 

1.0844 

.07780 

.92220 

15 

39 

4 

46 

. 8698 

. 1302 

.5841 

. 1968 

.3828 

.0845 

. 7791 

. 2209 

14 

56 

8 

47 

. 8725 

. 1275 

.5823 

. 2002 

.3808 

.0846 

. 7802 

. 2197 

13 

52 

12 

48 

. 8751 

. 1248 

.5805 

. 2036 

.3789 

.0847 

. 7814 

. 2186 

12 

48 

16 

49 

. 8778 

. 1222 

.5787 

2070 

.3770 

.0849 ■ 

. 7825 

. 2175 

11 

44 

20 

50 

.38805 

.61195 

2.5770 

.42105 

2.3750 

1.0850 

.07836 

.92164 

10 

40 

24 

51 

. 8832 

. 1168 

.5752 

. 2139 

.3731 

.0851 

. 7847 

. 2152 

9 

36 

28 

52 

. 8859 

. 1141 

.5734 

. 2173 

.3712 

.0853 

. 7859 

. 2141 

8 

32 

32 

53 

. 8886 

. 1114 

.5716 

. 2207 

.3692 

.0854 

. 7870 

. 2130 

7 

28 

36 

54 

. 8912 

. 1088 

.5699 

. 2242 

.3673 

.0855 

. 7881 

. 2118 

6 

24 

40 

55 

.38939 

.61061 

2.5681 

.42276 

2.3654 

1.0857 

.07893 

.92107 

5 

20 

44 

56 

. 8966 

. 1034 

.5663 

. 2310 

.3635 

.0858 

. 7904 

. 2096 

4 

16 

48 

57 

. 8993 

. 1007 

.5646 

. 2344 

.3616 

.0859 

. 7915 

. 2084 

3 

12 

52 

58 

. 9019 

. 0980 

.5628 

. 2379 

.3597 

.0861 

. 7927 

. 2073 

2 

8 

6(5 

59 

. 9046 

. 0954 

.5610 

. 24 i 3 

.3577 

.0862 

. 7938 

. 2062 

1 

4 

32 

60 

. 9073 

. 0927 

.5593 

. 2447 

.3558 

.0864 

. 7949 

. 2050 

0 

38 

M.S. 

M 1 

Cosine. 

Vrs.Sin.I 

Secame. 

Cotang. 

Tangent. 

Cosec’nt IVrs.Cos 

Sine, j 

M 

M.S. 

7 h 

112° 

€ 


Natural. 




67° 

4 h 

_! 



























232 Natural Lines. 


l h 

23° 

Natural Trigonometrical Functions. 

156° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

j Vrs.Sin 

Cosine. 

M 

M.S. 

3:3 

0 

.39073 

.60927 

2.5593 

.42447 

2.3558 

1.0864 

.07949 

.92050 

60 

28 

4 

l 

. 9100 

. 0900 

.5575 

. 2482 

.3539 

.0865 

. 7961 

. 2039 

69 

56 

8 

2 

. 9126 

. 0873 

.5558 

. 2516 

.3520 

.0866 

. 7972 

. 2028 

58 

52 

12 

3 

. 9153 

. 0846 

.5540 

. 2550 

.3501 

.0868 

. 7984 

. 2016 

57 

48 

16 

4 

. 9180 

. 0820 

.5523 

. 2585 

.3482 

.0869 

. 7995 

. 2005 

56 

44 

2!) 

5 

.39207 

.60793 

2.5506 

.42619 

2.3463 

1.0870 

.08006 

.91993 

55 

40 

24 

6 

. 9234 

. 0766 

.5488 

. 2654 

.3445 

.0872 

. 8018 

. 1982 

54 

36 

28 

7 

. 9260 

. 0739 

.5471 

. 2688 

.3426 

.0873 

. 8029 

. 1971 

53 

32 

32 

8 

. 9287 

. 0713 

.5453 

. 2722 

.3407 

.0874 

. 8041 

. 1959 

52 

23 

36 

9 

. 9314 

. 0686 

.5436 

. 5757 

.3388 

.0876 

. 8052 

. 1948 

51 

24 

40 

10 

.39341 

.60659 

2.5419 

.42791 

2.3369 

1.0877 

.08063 

.91936 

50 

20 

44 

11 

. 9367 

. 0632 

.5402 

. 2826 

.3350 

.0878 

. 8075 

. 1925 

49 

16 

48 

12 

. 9394 

. 0606 

.5:384 

. 2860 

.3332 

.0880 

. 8086 

. 1913 

48 

12 

52 

13 

. 9421 

. 0579 

.5367 

. 2894 

.3313 

.0881 

. 8098 

. 1902 

47 

8 

56 

14 

. 9448 

. 0552 

.5350 

. 2929 

.3294 

.0882 

. 8109 

. 1891 

46 

4 

33 

15 

.39474 

.60526 

2.5333 

.42963 

2.3276 

1.0884 

.08121 

.91879 

45 

27 

4 

16 

. 9501 

. 0499 

.5316 

. 2998 

.3257 

.0885 

. 8132 

. 1868 

44 

56 

8 

17 

. 9528 

. 0452 

.5299 

. 3032 

.3238 

.0886 

. 8144 

. 1856 

43 

52 

12 

18 

. 9554 

. 0445 

.5281 

. 3067 

.3220 

.0888 

. 8155 

. 1845 

42 

48 

16 

19 

. 9581 

. 0419 

.5264 

. 3101 

.3201 

.0889 

. 8167 

. 1833 

41 

44 

20 

20 

.39608 

.60392 

2.5247 

.43136 

2.3183 

1.0891 

.08178 

.91822 

40 

40 

24 

21 

. 9635 

. 0365 

.5230 

. 3170 

.3164 

.0892 

. 8190 

. 1810 

39 

36 

28 

22 

. 9661 

. 0339 

.5213 

. 3205 

.3145 

.0893 

. 8201 

. 1798 

38 

32 

32 

23 

. 9688 

. 0312 

.5196 

. 3239 

.3127 

.0895 

. 8213 

. 1787 

37 

28 

36 

24 

. 9715 

. 0285 

.5179 

. 3274 

.3109 

.0896 

. 8224 

. 1775 

36 

24 

40 

25 

.39741 

.60258 

2.5163 

.43308 

2.3090 

1.0897 

.08236 

.91764 

35 

20 

44 

26 

. 9768 

. 0232 

.5146 

. 3343 

.3072 

,08£>9 

. 8248 

. 1752 

34 

16 

48 

27 

. 9795 

. 0205 

.5129 

. 3377 

.3053 

.0900 

. 8259 

1741 

33 

12 

52 

28 

. 9821 

. 0178 

.5112 

. 3412 

.3035 

.0902 

. 8271 

. 1729 

32 

8 

56 

29 

. 9848 

. 0152 

.5095 

. 3447 

.3017 

.0903 

. 8282 

. 1718 

31 

4 

3 3 

30 

.39875 

.60125 

2.5078 

.43481 

2.2998 

1.0904 

.08294 

.91706 

30 

26 

4 

31 

. 9901 

. 0098 

.5062 

. 3516 

.2980 

.0906 

. 8306 

. 1694 

29 

56 

8 

32 

. 9928 

. 0072 

.5045 

. 3550 

.2962 

.0907 

. 8317 

. 1683 

28 

52 

12 

33 

. 9955 

. 0045 

.5028 

. 3585 

.2944 

.0908 

. 8329 

. 1671 

27 

48 

16 

34 

. 9981 

. 0018 

.5011 

. 3620 

.2925 

.0910 

. 8340 

. 1659 

26 

44 

20 

35 

.40008 

.59992 

2.4995 

.43654 

2.2907 

1.0911 

.08352 

.91648 

25 

40 

24 

36 

. 0035 

. 9965 

.4978 

. 3689 

.2889 

.0913 

. 8364 

. 1636 

24 

36 

28 

37 

. 0061 

. 9938 

.4961 

. 3723 

.2871 

.0914 

. 8375 

. 1625 

23 

32 

32 

38 

. 0088 

. 9912 

.4945 

. 3758 

.2853 

.0915 

. 8387 

. 1613 

22 

28 

36 

39 

. 0115 

. 9S85 

.4928 

. 3793 

.2835 

.0917 

. 8399 

. 1601 

21 

24 

40 

40 

.40141 

.59858 

2.4912 

.43827 

2.2817 

1.0918 

.08410 

.91590 

20 

20 

44 

41 

. 0168 

. 9832 

.4895 

. 3862 

.2799 

.0920 

. 8422 

. 1578 

19 

16 

48 

42 

. 0195 

. 9805 

.4879 

. 3897 

.2781 

.0921 

. 8434 

. 1566 

18 

12 

52 

43 

. 0221 

. 9778 

.4862 

. 3932 

.2763 

.0922 

. 8445 

. 1554 

17 

8 

56 

44 

. 0248 

. 9752 

.4846 

. 3966 

.2745 

.0924 

. 8157 

. 1543 

16 

4 

35 

45 

.40275 

.59725 

2.4829 

.44001 

2.2727 

1.0925 

.08469 

.91531 

15 

25 

4 

46 

. 0301 

. 9699 

.4813 

. 4036 

.2709 

.0927 

. 8480 

. 1519 

14 

66 

8 

47 

. 3328 

. 9672 

.4797 

. 4070 

.2691 

.0928 

. 8492 

. 1508 

13 

52 

12 

48 

. 0354 

. 9645 

.4780 

. 4105 

.2673 

.0929 

. 8504 

. 1496 

12 

48 

16 

49 

. 0381 

. 9619 

.4764 

. 4140 

.2655 

.0931 

. 8516 

. 1484 

11 

44 

20 

50 

.40408 

.59592 

2.4748 

.44175 

2.2637 

1.0932 

.08527 

.91472 

10 

40 

24 

51 

. 0434 

. 9566 

.4731 

. 4209 

.2619 

.0934 

. 8539 

. 1461 

9 

36 

28 

52 

. 0461 

. 9539 

.4715 

. 4244 

.2602 

.0935 

. 8551 

. 1449 

8 

32 

32 

53 

. 0487 

. 9512 

.4699 

. 4279 

.2584 

.0936 

. 8563 

. 1437 

7 

28 

36 

54 

. 0514 

. 94S6 

.4683 

. 4314 

.2566 

.0938 

. 8575 

. 1425 

6 

24 

40 

55 

.40541 

.59469 

2.4666 

.44349 

2.2548 

1.0939 

.0S586 

.91414 

5 

20 

44 

56 

. 0567 

. 9433 

.4650 

. 4383 

.2531 

.0941 

. 8598 

. 1402 

4 

16 

48 

57 

. 0594 

. 9406 

.4634 

. 4418 

.2513 

.0942 

. 8610 

. 1390 

3 

12 

52 

58 

. 0620 

. 9379 

.4618 

. 4453 

.2495 

.0943 

. 8622 

. 1378 

2 

8 

56 

59 

. 0647 

. 9353 

.4602 

. 4488 

.2478 

.0945 

. 8634 

. 1366 

1 

4 

36 

60 

. 0674 

. 9326 

.4586 

. 4523 

.2460 

.0946 

. 8645 

. l: 54 

0 

244 

M.S. 

7 h 

M 

113 f 

Cosine. 

Vrs.Sin. j 

Secant e. 1 

Cotaug.'Tangent. 

Natural. 

Cosec'ut 1 

Vrs. Cos 1 

% 

Sine. 

M 

66 ° 

M.S. 

4 h 



























Natural Lines. 


283 


l h 

24° 

Natui’al Trigonometrical Functions 

155° 

10 h 

M.S. 

M 

1 Sine. 

Vrs.Cos. 

ICosee’nte 

Tang. 

Cotang. 

Secante 

jVrs.Sin 

Cosine. 

1 M 

M.S 

36 

0 

.40674 

.59326 

2.4586 

.44523 

2.2460 

1.0946 

1.08645 

.91354 

60 

21 

4 

1 

. 0700 

. 9300 

j .4570 

. 4558 

.2443 

.0948 

. 8657 

. 1343 

59 

56 

8 

2 

. 0727 

. 9273 

.4554 

. 4593 

.2425 

.0949 

. 8669 

. 1331 

58 

52 

12 

3 

. 0753 

. 9247 

.4538 

. 4627 

.2408 

.0951 

. 8681 

. 1319 

57 

48 

16 

4 

. 0780 

. 9220 

.4522 

. 4662 

.2390 

.0952 

. 8693 

. 1307 

56 

44 

20 

5 

.40806 

.59193 

2.45U6 

.44697 

2.2373 

1.0953 

.08705 

.91295 

55 

40 

24 

6 

. 0833 

. 9167 

.-1490 

. 4732 

.2355 

.0955 

. 8716 

. 1283 

54 

36 

28 

7 

. 0860 

. 9140 

.4474 

. 4767 

.2338 

.0956 

. 8728 

. 1271 

53 

32 

32 

8 

. 0886 

. 9114 

.4458 

. 4802 

.2320 

.0958 

. 8740 

. 1260 

52 

28 

36 

9 

. 0913 

. 9087 

.4442 

. 4837 

.2303 

.0959 

. 8752 

. 1248 

51 

24 

40 

10 

.40939 

59061 

2.4426 

.44872 

2.2286 

1.0961 

.08764 

.91236 

50 

20 

44 

11 

. 0966 

. 9034 

| .4411 

. 4907 

.2268 

.0962 

. 8776 

. 1224 

49 

16 

48 

12 

. 0992 

. 9008 

* .4395 

. 4942 

.2251 

.0963 

. 8788 

. 1212 

48 

12 | 

52 

13 

. 1019 

. 8981 

, .4379 

. 4977 

.2234 

.0965 

. 8800 

. 1200 

47 

8 

56 

14 

. 1045 

. 8955 

.4363 

. 5012 

.2216 

.0966 

. 8812 

. 1188 

46 

4 

37 

15 

.41072 

.58928 

2.4347 

.45047 

2.2199 

1.0968 

.08824 

.91176 

45 

23 

4 

16 

. 1098 

. 8901 

1 .4332 

. 5082 

.2182 

.0969 

. 8836 

. 1164 

44 

56 

8 

17 

. 1125 

. 8875 

.4316 

. 5117 

.2165 

.0971 

. 8848 

. 1152 

43 

52 

12 

18 

. 1151 

. 8848 

.4300 

. 5152 

.2147 

.0972 

. 8860 

. 1140 

42 

48 

16 

19 

. 1178 

. 8822 

.4285 

. 5187 

.2130 

.0973 

. 8872 

. 1128 

41 

44 

20 

20 

.41204 

.58795 

2.4269 

.45222 

2.2113 

1.0975 

.08884 

.91116 

40 

40 

24 

21 

. 1231 

. 8769 

.4254 

. 5257 

.2096 

.0976 

. 8896 

. 1104 

39 

36 

28 

22 

. 1257 

. 8742 

.4238 

. 5292 

.2079 

.0978 

. 8908 

. 1092 

38 

32 

32 

23 

. 1284 

. 8716 

.4222 

. £327 

.2062 

.0979 

. 8920 

. 1080 

37 

28 

36 

24 

. 1310 

. 8689 

.4207 

. 5362 

.2045 

.0981 

. 8932 

. 1068 

36 

24 

40 

25 

.41337 

.58663 

2.4191 

.45397 

2.2028 

1.0982 

.08944 

.91056 

35 

20 

44 

26 

. 1363 

. 8636 

.4176 

. 5432 

.2011 

.0984 

. 8956 

. 1044 

34 

16 

48 

27 

. 1390 

. 8610 

.4160 

. 6467 

.1994 

.0985 

. 8968 

. 1032 

33 

12 , 

52 

28 

. 1416 

. 8584 

.4145 

. 5502 

.1977 

.0986 

. 8980 

. 1020 

32 

8 

56 

29 

. 1443 

. 8557 

.4130 

. 5537 

.1960 

.0988 

. 8992 

. 1008 

31 

4 ! 

38 

30 

.41469 

.58531 

2.4114 

.45573 

2.1943 

1.0989 

.09004 

.90996 

30 

22 

4 

31 

. 1496 

. 8501 

.4099 

. 5608 

.1926 

.0991 

. 9016 

. 0984 

29 

56 

8 

32 

. 1522 

. 8478 

.4083 

. 5643 

.1909 

.0992 

. 9028 

. 0972 

28 

52 

12 

33 

. 1549 

. 8451 

.4068 

. 5678 

.1892 

.0994 

. 9040 

. 0960 

27 

48 

16 

34 

. 1575 

. 8425 

.4053 

. 5713 

.1875 

.0995 

. 9052 

. 0948 

26 

44 

20 

35 

.41602 

.58398 

2.4037 

.45748 

2.1859 

1.0997 

.09064 

.90936 

25 

40 

24 

36 

. 1628 

. 8372 

.4022 

. 5783 

.1842 

.0998 

. 9076 

. 0924 

24 

36 

28 

37 

. 1654 

. 8345 

.4007 

. 5819 

.1825 

.1000 

. 9088 

. 0911 

23 

32 

32 

38 

. 1681 

. 8319 

.3992 

. 5854 

.1808 

.1001 

. 9101 

. 0899 

22 

28 

36 

39 

. 1707 

. 8292 

.3976 

. 5889 

.1792 

.1003 

. 9113 

. 0887 

21 

24 

40 

40 

.41734 

.58266 

2.3961 

.45924 

2.1775 

1.1004 

.09125 

.90875 

20 

20 

44 

41 

. 1760 

. 8240 

.3946 

. 5960 

.1758 

.1005 

. 9137 

. 0863 

19 

16 

48 

42 

. 1787 

. 8213 

.3931 

. 5995 

.1741 

.1007 

. 9149 

. 0851 

18 

12 

52 

43 

. 1813 

. 8187 

.3916 

. 6030 

.1725 

.1008 

. 9161 

. 0839 

17 

8 

56 

44 

. 1839 

. 8160 

.3901 

. 6065 

.1708 

.1010 

. 9173 

. 0826 

16 

4 

39 

45 

.41866 

.58134 

2.3886 

.46101 

2.1692 

1.1011 

.09186 

.90814 

15 

21 

4 

46 

. 1892 

. 8108 

.3871 

. 6136 

.1675 

.1013 

. 9198 

. 0802 

14 

56 

8 

47 

. 1919 

. 8081 

.3856 

. 6171 

.1658 

.1014 

. 9210 

. 0790 

13 

52 

12 

48 

. 1945 

. 8055 

.3841 

. 6206 

.1642 

.1016 

. 9222 

. 0778 

12 

48 

16 

49 

. 1972 

. 8028 

.3826 

. 6242 

.1625 

.1017 

. 9234 

. 0765 

11 

44 

20 

50 

.41998 

.58002 

2.3811 

.46277 

2.1609 

1.1019 

.09247 

,90753 

10 

40 

24 

51 

. 2024 

. 7975 

.3796 

. 6312 

.1592 

.1020 

. 9259 

. 0741 

9 

36 

28 

52 

. 2051 

. 7949 

.3781 

. 6348 

.1576 

.1022 

. 9271 

. 0729 

8 

32 

32 

53 

. 2077 

. 7923 

.8766 

. 6383 

.1559 

.1023 

. 9283 

. 0717 

7 

28 

36 

54 

. 2103 

. 7896 

.3751 

. 6418 

.1543 

.1025 

. 9296 

. 0704 

6 

24 

40 

55 

.42130 

.57870 

2.3736 

.46454 

2.1527 

1.1026 

.09308 

.90692 

5 

20 

44 

56 

. 2156 

. 7844 ! 

.3721 

. 6489 

.1510 

.1028 | 

. 9320 

. 0680 

4 

16 

48 

57 

. 2183 

. 7817 

.3706 

. 6524 

.1494 

.1029 | 

. 9332 

. 0668 

3 

12 

52 

58 

. 2209 

. 7791 

.3691 

. 6560 

.1478 

.1031 1 

. 9345 

. 0655 

2 

8 

56 

59 

. 2235 

. 7764 

.3677 

. 6595 

.1461 

.1032 

. 9357 

. 0643 

1 

4 

40 

60 

. 2262 

. 7738 

.8662 

. 6631 

.1445 

.1034 I 

. 9369 

. 0631 

0 

20 

M. S.j 

M ! 

Cosine. , 

Vra.Sin.J 

Secante. 

Cotaug.'Tangent. 

Cosec'ut [Vrs.Cos 

Sine. 

M 

M.S. 

7 h i 

114 C 




Natural. 




4 h 


























































234 Natural Lines. 


l h 

25 c 

Natural Trigonometrical Functions 

154° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

|Vrs. Sin 

I Cosine. 

M 

M.S. 

40 

0 

.42262 

.57738 

2.3662 

.46631 

2.1445 

1.1034 

.09369 

.90631 

60 

30 

4 

1 

. 2288 

. 7712 

.3647 

. 6666 

.1429 

.1035 

. 9381 

. 0618 

59 

56 

8 

2 

. 2314 

. 7685 

.3632 

. 6702 

.1412 

.1037 

. 9394 

. 0606 

58 

52 

12 

3 

. 2341 

. 7659 

.3618 

. 6737 

.1396 

.1038 

. 9406 

. 0594 

57 

48 

16 

4 

. 2367 

. 7633 

.3603 

. 6772 

.1380 

, 1040 

. 9418 

. 0581 

56 

44 

20 

5 

.42394 

.57606 

2.3588 

.46808 

2.1364 

1.1041 

.09431 

.90569 

55 

40 

24 

6 

. 2420 

. 7580 

.3574 

. 6843 

.1348 

.1043 

. 9443 

. 0557 

54 

36 

28 

7 

. 2446 

. 7554 

.3559 

. 6879 

.1331 

.1044 

. 9455 

. 0544 

53 

32 

32 

8 

. 2473 

. 7527 

.3544 

. 6914 

.1315 

.1046 

. 9468 

. 0532 

52 

28 

36 

9 

. 2499 

. 7501 

.3530 

. 6950 

.1299 

.1047 

. 9480 

. 052U 

51 

24 

40 

10 

.42525 

.57475 

2.3515 

.46985 

2.1283 

1.1049 

09492 

.90507 

50 

20 

44 

11 

. 2552 

. 7448 

.3501 

. 7021 

.1267 

.1050 

. 9505 

. 0495 

49 

10 i 

48 

12 

. 2578 

. 7422 

.3486 

. 7056 

.1251 

.1052 

. 9517 

. 0483 

48 

12 

52 

13 

. 2604 

. 7396 

.3472 

. 7092 

.1235 

.1053 

. 9530 

. 0470 

47 

8 

56 

14 

. 2630 

. 7369 

.3457 

. 7127 

.1219 

.1055 

. 9542 

. 0458 

46 

4 

41 

15 

.42657 

.57343 

2.3443 

.47163 

2.1203 

1.1056 

.09554 

.90445 

45 

19 

4 

16 

. 2683 

. 7317 

.3428 

. 7199 

.1187 

.1058 

. 9567 

. 0433 

44 

56 

8 

17 

. 2709 

. 7290 

.3414 

. 7234 

.1171 

.1059 

. 9579 

. 0421 

43 

52 

12 

18 

. 2736 

. 7264 

.3399 

. 7270 

.1155 

.1061 

. 9592 

. 0408 

42 

48 

16 

19 

. 2762 

7238 

.3385 

. 7305 

.1139 

.1062 

. 9604 

. 0396 

41 

44 

20 

20 

.42788 

.57212 

2.3371 

.47341 

2.1123 

1.1064 

.09617 

.90383 

40 

40 

24 

21 

. 2815 

. 7185 

.3356 

. 7376 

.1107 

.1065 

. 9629 

. 0371 

39 

36 

28 

22 

. 2841 

. 7159 

.3342 

. 7412 

.1092 

.1067 

. 9641 

. 0358 

38 

32 

32 

23 

. 2867 

. 7133 

.3328 

. 7448 

.1076 

.1068 

. 9654 

. 0346 

37 

28 

36 

24 

. 2893 

. 7106 

.3313 

. 7483 

.1060 

.1070 

. 9666 

. 0333 

36 

24 

40 

25 

.42920 

.57080 

2.3299 

.47519 

2.1044 

1.1072 

.09679 

.90321 

35 

20 

44 

26 

. 2946 

. 7054 

.3285 

. 7555 

.1028 

.1073 

. 9691 

. 0308 

34 

16 

48 

27 

. 2972 

. 7028 

.3271 

. 7590 

.1013 

.1075 

. 9704 

. 0296 

33 

12 

52 

28 

. 2998 

. 7001 

.3256 

. 7626 

.0997 

.1076 

. 9716 

. 0283 

32 

8 

56 

29 

. 3025 

. 6975 

.3242 

. 7662 

.0981 

.1078 

. 9729 

. 0271 

31 

4 

4:3 

30 

.43051 

.56949 

2.3228 

.47697 

2.0965 

1.1079 

.09741 

.90258 

30 

18 

4 

31 

. 3077 

. 6923 

.3214 

. 7733 

.0950 

.1081 

. 9754 

. 0246 

29 

56 

8 

32 

. 3104 

. 6896 

.3200 

. 7769 

.0934 

.1082 

. 9766 

. 0233 

28 

52 

12 

33 

. 3130 

. 6870 

.3186 

. 7805 

.0918 

.1084 

. 9779 

. 0221 

27 

48 

16 

34 

. 3156 

. 6844 

.3172 

. 7840 

.0903 

.1085 

. 9792 

. 0208 

26 

44 

20 

35 

.43182 

.56818 

2.3158 

.47876 

2.0887 

1.1087 

.09804 

.90196 

25 

40 

24 

36 

. 3208 

. 6791 

.3143 

. 7912 

.0872 

.1088 

. 9817 

. 0183 

2-1 

36 

28 

37 

. 3235 

. 6765 

.3129 

. 7948 

.0856 

.1090 

. 9829 

. 0171 

23 

32 

32 

38 

. 3261 

. 6739 

.3115 

. 7983 

.0840 

.1092 

. 9842 

. 0158 

22 

28 

36 

39 

. 3287 

. 6713 

.3101 

. 8019 

.0825 

.1093 

. 9854 

. 0145 

21 

24 

40 

40 

.43313 

.56686 

2.3087 

.48055 

2.0809 

1.1095 

.09867 

.90133 

20 

20 

44 

41 

. 3340 

. 6660 

.3073 

. 8091 

.0794 

.1096 

. 9880 

. 0120 

19 

16 

48 

42 

. 3366 

. 6634 

.3059 

. 8127 

.0778 

.1098 

. 9892 

. 0108 

18 

12 

52 

43 

. 3392 

. 6603 

.3046 

. 8162 

.0763 

.1099 

. 9905 

. 0095 

17 

8 

56 

44 

. 3418 

. 6582 

.3032 

. 8198 

.0747 

.1101 

. 9917 

. 0082 

16 

4 

43 

45 

.43444 

.66555 

2.3018 

.48234 

2.0732 

1.1102 

.09930 

.90070 

15 

17 

4 

46 

. 3471 

. 6529 

.3004 

. 8270 

.0717 

.1104 

. 9943 

. 0057 

14 

56 

8 

47 

. 3497 

. 6503 

.2990 

. 8306 

.0701 

.1106 

. 9955 

. 0044 

13 

52 

12 

48 

. 3523 

. 6477 

.2976 

. 8342 

.0686 

.1107 

. 9968 

. 0032 

12 

48 

16 

49 

. 3549 

. 6451 

.2962 

8378 

.0671 

.1109 

. 9981 

. 0019 

11 

44 

20 

50 

.43575 

.56424 

2.2949 

.48414 

2.0655 

1.1110 

.09993 

.90006 

10 

40 

24 

51 

. 3602 

. 6398 

.2935 

. 8449 

.0640 

.1112 

.10006 

.89994 

9 

36 

28 

52 

. 3628 

. 6372 

.2921 

. 8485 

.0625 

.1113 

. 0019 

. 9981 

8 

32 

32 

53 

. 3654 

. 6346 

.2907 

. 8521 

.0609 

.1115 

. 0031 

. 9968 

7 

28 

36 

54 

7 3680 

. 6320 

.2894 

. 8557 

.0594 

.1116 

. 0044 

. 9956 

6 

24 

40 

55 

.43706 

.56294 

2.2880 

.48593 

2.0579 

1.1118 

.10057 

.89943 

5 

20 

44 

56 

. 3732 

. 6267 

.2866 

. 8629 

.0564 

.1120 

. 0070 

. 9930 

4 

16 

48 

57 

. 3759 

. 6241 

.2853 

. 8665 

.0548 

.1121 

. 0082 

. 9918 

3 

12 

52 

58 

. 3785 

. 6215 

.2839 

. 8701 

.0533 

.1123 

. 0095 

. 9905 

2 

8 

56 

59 

. 3811 

. 6189 

.2825 

. 8737 

.0518 

.1124 

. 0108 

. 9892 

1 

4 

44 

60 

. 3837 

. 6163 

.2812 

. 8773 

.0503 

.1126 

. 0121 

. 9879 

0 

16 

M.S. 

M 

Cosine. 

Yrs.Sin.I Secante. 

Cotaug.’Tangent. 

Cosec’nt 

Vrs.Cos 

Sine. 

M 

M.S. 

7 h 

115 

D 



Natural. 




64° 

4 h 


























Natural Lines. 


235 


Natural Trigonometrical Functions. 153° 10 h 


M.S. 

m 

Sine. 

VTs.Cos. 

Cosec'nte 

Tang. 

Cotaug. 

Secante. ,Vrs. Sin 

Cosine. 

M 

M.S. 

11 

0 

.43837 

.56163 

2.2812 

.48773 

2.0503 

1.1126 

.10121 

•89879 

60 

16 

4 

l 

. 3863 

. 6137 

.2798 

. 88U9 

.0488 

.1127 

. 0133 

. 9867 

59 

56 

8 

2 

. 3889 

. 6111 

.2784 

. 8845 

.0473 

.1129 

. 0146 

. 9854 

58 

52 

12 

3 

. 3915 

. 6084 

.2771 

. 8881 

.0458 

.1131 

. 0159 

. 9841 

57 

48 

16 

4 

. 3942 

. 6058 

.2757 

. 8917 

.0443 

.1132 

. 0172 

. 9828 

56 

44 

20 

5 

.43968 

.56032 

2.2744 

.48953 

2.0427 

1.1134 

.10184 

.89815 

55 

40 

24 

6 

. 3994 

. 6006 

.2730 

. 8989 

.0412 

.1135 

. 0197 

. 9303 

54 

36 

28 

7 

. 4020 

. 5980 

.2717 

. 9025 

.0397 

.1137 

. 0210 

. 9790 

53 

32 

32 

8 

. 4046 

. 5954 

.2703 

. 9062 

.0382 

.1139 

. 0223 

. 9777 

52 

28 

36 

9 

. 4072 

. 5928 

.2690 

. 9098 

.0367 

.1140 

. 0236 

. 9764 

51 

24 

40 

10 

.44098 

.55902 

2.2676 

.49134 

2.0352 

1.1142 

.10218 

.i-9751 

50 

20 

44 

11 

. 4124 

. 5875 

.2663 

. 9170 

.0338 

.1143 

. 0261 

. 9739 

49 

16 

48 

12 

. 4150 

. 5849 

.2650 

. 9206 

.0323 

.1145 

. 0274 

. 9726 

48 

12 

52 

13 

. 4177 

. 5823 

.2636 

. 9242 

.0308 

.1147 

. 0287 

. 9713 

47 

8 

66 

14 

. 4203 

. 5797 

.2623 

. 9278 

.0293 

.1148 

. 0300 

. 9700 

46 

4 

45 

15 

.44229 

.55771 

2.2610 

.49314 

2.0278 

1.1150 

10313 

.89687 

45 

15 

4 

16 

. 4255 

. 5745 

.2596 

. 9351 

.0263 

.1151 

0326 

. 9674 

44 

56 

8 

17 

. 4281 

. 5719 

.2583 

. 9387 

.0248 

.1153 

. 0338 

. 9661 

43 

52 

12 

IS 

. 4307 

. 5693 

.2570 

. 9123 

.0233 

.1155 

. 0351 

. 9649 

42 

48 

16 

19 

. 4333 

. 5667 

.2556 

. 9159 

.0219 

.1156 

. 0364 

. 9636 

41 

44 

20 

20 

.44359 

.55611 

2.2543 

.49195 

2.0204 

1.1158 

.10377 

.89623 

40 

40 

24 

21 

. 4385 

. 5615 

.2530 

. 9532 

.0189 

.1159 

. 0390 

. 9610 

39 

36 

28 

22 

. 4411 

. 55S9 

.2517 

. 9568 

.0174 

.1161 

. 0403 

. 9597 

38 

32 

32 

23 

. 4437 

. 5562 

.2503 

. 9601 

.0159 

.1163 

0416 

. 95X4 

37 

28 

36 

24 

. 4463 

. 5536 

.2490 

. 9640 

.0145 

.1164 

. 0429 

. 9571 

36 

24 

40 

25 

.44489 

.55510 

2.2477 

.49677 

2.0130 

1.1166 

.10442 

.89558 

35 

20 

44 

26 

. 1516 

. 5484 

.2461 

. 9713 

.0115 

.1167 

. 0455 

. 9545 

34 

16 

48 

27 

. 4542 

. 5458 

.2451 

. 9749 

.0101 

.1169 

. 0468 

. 9532 

o3 

12 

52 

28 

. 456S 

. 5432 

.2438 

. 9785 

.0086 

.1171 

. 0481 

. 9519 

32 

8 

56 

29 

. 4594 

. 5406 

.2425 

. 9822 

.0071 

.1172 

. 0493 

. 95U6 

31 

4 

46 

30 

.44620 

.55380 

2.2411 

.49858 

2.0057 

1.1174 

.10506 

.89493 

30 

14 

4 

31 

. 4646 

. 5354 

.2398 

. 9894 

.0042 

.1176 

. 0519 

. 94X0 

29 

56 

8 

32 

. 4672 

. 5328 

.2385 

. 9931 

.0028 

.1177 

. 0532 

. 9467 

28 

52 

12 

33 

. 4698 

. 5302 

.2372 

. 9967 

.0013 

.1179 

. 0515 

. 9154 

27 

48 

16 

34 

. 4724 

. 5276 

.2359 

.50093 

1.9998 

.1180 

. 0553 

. 9441 

26 

44 

20 

35 

.44750 

.55250 

2.2346 

.50010 

1.9981 

1.1182 

.10571 

.89428 

25 

40 

24 

36 

. 4776 

. 5221 

.2333 

. 0076 

.6969 

.1184 

. 0584 

. 9415 

24 

36 

28 

37 

. 4S02 

. 5198 

.2320 

. 0113 

.9955 

.1185 

. 0598 

. 9402 

23 

32 

32 

38 

. 4828 

. 5172 

.2307 

. 0149 

.9940 

.1187 

. 0611 

. 9389 

22 

28 

36 

39 

. 4854 

. 5116 

.2294 

. 0185 

.9926 

.1189 

. 0624 

. 1376 

21 

24 

40 

40 

.44880 

.55120 

2.2282 

.50222 

1.9912 

1.1190 

.10637 

.89363 

20 

20 

44 

41 

. 4906 

. 5094 

.2269 

. 0258 

.9897 

.1192 

. 0650 

. 9350 

19 

16 

48 

42 

. 4932 

. 5068 

.2256 

. 0295 

.9883 

.1193 

. 0663 

. 9337 

18 

12 

52 

43 

. 4958 

. 5042 

.2243 

. 0331 

. .9868 

.1195 

. 0676 

. 9324 

17 

8 

56 

44 

. 4984 

. 5016 

.2230 

. 0368 

.9854 

.1197 

. 0689 

. 9311 

16 

4 

47 

45 

.45010 

51990 

2.2217 

.50401 

1.9840 

1.1198 

.10702 

.89298 

15 

13 

4 

46 

. 5036 

. 4964 

.2204 

. 0141 

.9825 

.1200 

. 0"'5 

. 9285 

14 

56 

8 

47 

. 5062 

. 4938 

.2192 

. 0477 

.9811 

.1202 

. 0728 

. 9272 

13 

52 

12 

48 

. 5088 

. 4912 

.2179 

. 0511 

.9797 

.1203 

. 0741 

. 9258 

12 

48 

16 

49 

. 5114 

. 4886 

.2166 

. 0550 

.9782 

.1205 

. 0754 

. 9245 

11 

44 

20 

50 

.45140 

.54860 

2.2153 

.50587 

1.9768 

1.1207 

.10768 

.89232 

10 

40 

24 

51 

. 5166 

. 4834 

.2141 

. 0623 

.9754 

.1208 

. 0781 

. 9219 

9 

36 

28 

52 

. 5191 

. 4808 

.2128 

. 0600 

.9739 

.1210 

. 0794 

. 9206 

8 

32 

32 

53 

. 5217 

. 4782 

.2115 

. 0696 

.9725 

.1212 

. 0807 

. 9193 

7 

23 

36 

54 

. 5243 

. 4756 

.2103 

. 0733 

.9711 

.1213 

. 0820 

. 9180 

6 

24 

40 

55 

.45269 

.54730 

2.2090 

.50769 

1.9697 

1.1215 

.16833 

.89166 

5 

20 

44 

56 

. 5295 

. 4705 

.2077 

. 0806 

.9683 

.1217 

. 0846 

. 9153 

4 

16 

48 

57 

. 5321 

. 4679 

.2065 

. 0843 

.9668 

.1218 

. 0860 

. 9140 

3 

12 

52 

58 

. 5347 

. 4653 

.2052 

. 0879 

.9654 

.1220 

. 0873 

. 9127 

2 

8 

56 

59 

. 5373 

. 4627 

.2039 

. 0916 

.964) 

.1222 

. 0886 

. 9114 

1 

4 

48 

60 

. 5399 

. 4601 

.2027 

. 0952 

.9626 

.1223 

. 0899 

. 9101 

0 

1)1 

M. S. 

M 

Cosine. 

Vrs. Sin. 

Secauto. 

Cotang. 

Tangent. 1 

Cosec’nt A’rs. Cos 

Sine. 

M 

M.S. 

7 h !l 16 v 




Natural. 




63° 

4 b 




































s&G Natural Linus. 


l h 

27° 

Natural Trigonometrical Functions 

152° 

10 h 

M. S. 

M 

Sine. 

Yrs.Cos. 

Cosec’nte 

Tang. 

| Cotaug. 

Secante. 

Vrs. Sin 

Cosine. J M 

M. S. 

‘J 8 

0 

.45399 

.54601 

2.2027 

.50952 

1.9626 

1.1223 

.10899 

.89101 

60 

12 

4 

1 

. 5425 

. 4575 

.2014 

. 0989 

.9612 

.1225 

. 0912 

. 9087 

69 

56 

8 

2 

. 5451 

. 4549 

.2002 

. 1026 

.9598 

.1226 

. 0926 

. 9074 

68 

52 

12 

3 

. 5477 

. 4523 

.1989 

. 1062 

.9584 

.1228 

. 0939 

. 9061 

57 

48 

16 

4 

. 5503 

. 4497 

.1977 

. 1099 

.9570 

.1230 

. 0952 

. 9048 

56 

44 

20 

5 

.45528 

.54471 

2.1964 

.51136 

1.9556 

1.1231 

.10965 

.89034 

55 

40 

24 

6 

. 6554 

. 4445 

.1952 

. 1172 

.9542 

.1233 

. 0979 

. 9021 

54 

36 

28 

7 

. 55S0 

. 4420 

.1939 

. 1209 

.9528 

.1235 

. 0992 

. 9008 

53 

32 

32 

8 

. 6606 

. 4394 

.1927 

. 1246 

.9514 

.1237 

. 1005 

. 8995 

52 

28 

36 

9 

. 5632 

. 4368 

.1914 

. 1283 

.9500 

.1238 

. 1018 

. 8981 

61 

24 

40 

10 

.45658 

.54342 

2.1902 

.51319 

1.9486 

1.1240 

.11032 

.88968 

50 

2o 

44 

11 

. 5684 

. 4316 

.1889 

. 1356 

.9472 

.1242 

. 1045 

. 8955 

49 

16 

48 

12 

. 5710 

. 4290 

.1877 

. 1393 

.9458 

.1243 

. 1058 

. 8942 

48 

12 

52 

13 

. 5736 

. 4264 

.1865 

. 1430 

.9444 

.1245 

. 1072 

. 8928 

47 

8 

56 

14 

. 5761 

. 4238 

.1852 

. 1466 

.9430 

.1247 

. 1085 

. 8915 

46 

4 

49 

15 

.45787 

.54213 

2.1840 

.51503 

1.9416 

1.1248 

.11098 

.88902 

45 

11 

4 

16 

. 5813 

. 4187 

.1828 

. 1540 

.9402 

.1250 

. 1112 

. 8888 

44 

56 

8 

17 

. 5839 

. 4161 

.1815 

. 1577 

.9388 

.1252 

. 1125 

. 8875 

43 

52 

12 

18 

. 5865 

. 4135 

.1803 

. 1614 

.9375 

.1253 

1138 

. 8862 

42 

48 

16 

19 

. 5891 

. 4109 

.1791 

. 1651 

.9361 

.1255 

. 1152 

. 8848 

41 

41 

20 

20 

.45917 

.54083 

2.1778 

.51687 

1.9347 

1.1257 

.11165 

.88835 

40 

40 

24 

21 

. 5942 

. 4057 

.1766 

. 1724 

.9333 

.1258 

. 1178 

. 8822 

39 

36 

28 

22 

. 5968 

. 4032 

.1754 

. 1761 

.9319 

.1260 

. 1192 

. 8808 

38 

32 

32 

23 

. 5994 

. 4006 

.1742 

. 1798 

.9306 

.1262 

. 1205 

. 8795 

37 

28 

36 

24 

. 6020 

. 3980 

.1730 

. 1835 

.9292 

.1264 

. 1218 

. 8781 

36 

24 

40 

25 

.46046 

.53954 

2.1717 

.51872 

1.9278 

1.1265 

.11232 

.88768 

35 

20 

41 

26 

. 6072 

. 3928 

.1705 

. 1909 

.9264 

.1267 

. 1245 

. 8755 

34 

16 

48 

27 

. 6097 

. 3902 

.1693 

. 1946 

.9251 

.1269 

. 1259 

. 8741 

33 

12 

52 

28 

. 6123 

. 3877 

.1681 

. 1983 

.9237 

.1270 

. 1272 

. 8728 

32 

8 

56 

29 

. 6149 

. 3851 

.1669 

. 2020 

.9223 

.1272 

. 1285 

. 8714 

31 

4 

50 

30 

.46175 

.53825 

2.1657 

.52057 

1.9210 

1.1274 

.11299 

.88701 

30 

10 

4 

31 

. 6201 

. 3799 

.1645 

. 2094 

.9196 

.1275 

. 1312 

. 8688 

29 

56 

8 

32 

. 6226 

. 3773 

.1633 

. 2131 

.9182 

.1277 

. 1326 

. 8674 

28 

52 

12 

33 

. 6252 

. 3748 

.1620 

. 2168 

.9169 

.1279 

. 1339 

. 8661 

27 

48 

16 

34 

. 6278 

. 3722 

.1608 

. 2205 

.9155 

.1281 

. 1353 

. 8647 

26 

44 

20 

35 

.46304 

.53696 

2.1596 

.52242 

1.9112 

1.1282 

.11366 

.88634 

25 

40 

24 

36 

. 6330 

. 3670 

.1584 

. 2279 

.9128 

.1284 

. 1380 

. 8620 

24 

36 

28 

37 

. 6355 

. 3645 

.1572 

. 2316 

.9115 

.1286 

1393 

. 8607 

23 

32 

32 

38 

. 6381 

. 3619 

.1560 

. 2353 

.9101 

.1287 

. 1407 

. 8593 

22 

28 

36 

39 

• 6407 

. 3593 

.1548 

. 2390 

.9088 

.1289 

. 1420 

. 8580 

21 

24 

40 

40 

.46433 

.53567 

2.1536 

.52427 

1.9074 

1.1291 

.11434 

.88566 

20 

20 

44 

41 

. 6458 

. 3541 

.1525 

. 2464 

.9061 

.1293 

. 1447 

. 8553 

19 

16 

48 

42 

. 6484 

. 3516 

.1513 

. 2501 

.9047 

.1294 

. 1461 

. 8539 

18 

12 

52 

43 

. 6510 

. 3490 

.1501 

. 2536 

.9034 

.1296 

. 1474 

. 8526 

17 

8 

56 

44 

. 6536 

. 3464 

.1489 

. 2575 

.9020 

.1298 

. 1488 

. 8512 

16 

4 

51 

45 

.46561 

.53438 

2.1477 

.52612 

1.9007 

1.1299 

.11501 

.88499 

15 

9 

4 

46 

. 6587 

. 3413 

.1465 

. 2650 

.8993 

.1301 

. 1515 

. 8485 

14 

56 

8 

47 

• 6613 

. 3387 

.1453 

. 2687 

.8980 

.1303 

. 1528 

. 8472 

13 

52 

12 

48 

. 6639 

. 3361 

.1441 

. 2724 

.8967 

.1305 

. 1542 

. 8458 

12 

48 

16 

49 

• 6664 

. 3336 

.1430 

. 2761 

.8953 

.1306 

. 1555 

. 8444 

11 

44 

20 

50 

.46690 

.53310 

2.1418 

.52798 

1.8940 

1.1308 

.11569 

.88431 

10 

40 

24 

51 

• 6716 

. 3284 

.1406 

. 2836 

.8927 

.1310 

. 1583 

. 8417 

9 

36 

28 

52 

. 6741 

. 3258 

.1394 

. 2873 

.8913 

.1312 

. 1596 

. 8404 

8 

32 

32 

53 

. 6767 

. 3233 

.1382 

. 2910 

.8900 

.1313 

. 1610 

. 8390 

7 

28 

36 

54 

. 6793 

. 3207 

.1371 

. 2947 

.8887 

.1315 

. 1623 

. 8376 

6 

24 

40 

55 

.46819 

.53181 

2.1359 

.52984 

1.8873 

1.1317 

.11637 

.88363 

5 

20 

44 

56 

. 6S44 

. 3156 

.1347 

. 3022 

.8860 

.1319 

. 1651 

. 8349 

4 

16 

48 

57 

. 6870 

. 3L30 

.1335 

. 3059 

.8847 

.1320 

. 1664 

. 8336 

3 

12 

52 

58 

. 6896 

. 3104 

.1324 

. 3096 

.8834 

.1322 

. 1678 

. 8322 

2 

8 

56 

59 

. 6921 

. 3078 

.1312 

. 3134 

.8820 

.1324 

. 1691 

. 8308 

i 

4 

52 

60 

. 6947 

. 3053 

.1300 

. 3171 

.8807 

.1326 f 

. 1705 

. 8295 

0 

8 

M. S. 

M 

Cosine. 1 

Vrs. Sin. 

Secante. 

Cotang. 

Tangent. 

Cosec'nt Vrs. Cos 

Sine. 

M 

VI S. 

7 h 

117' 

D 



Natural. 




62°j 

4“ 

























Natural Lines. 


m 


lh- 

28° 

Natural Trigonometrical Functions 

151° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Oosec’nte 

Tang. 

j Cotaug. 

Secante. 

|Vrs. Sin 

Cosine. 

M 

M.S. 

5 2 

0 

.46947 

.53053 

2.1300 

.53171 

1.8807 

1.1326 

.11705 

.88295 

60 

8 

4 

1 

. 0j73 

. 3027 

.1289 

. 3208 

.8794 

.1327 

. 1719 

. 8281 

59 

56 

8 

2 

. 6998 

. 300i 

.127/ 

. 3245 

.8781 

.1329 

. 1732 

. 8267 

58 

52 

12 

3 

. 7021 

. 2976 

.1266 

. 3283 

.3768 

.1331 

. 1746 

. S254 

57 

43 

lfi 

4 

. 7050 

. 2950 

.1251 

. 3320 

.8754 

.1333 

. 1760 

. 8240 

56 

44 

20 

5 

.47075 

.52924 

2.1242 

.53358 

1.8741 

1.1334 

.11774 

.88226 

55 

40 

24 

6 

. 7101 

. 2899 

.1231 

. 3395 

.8728 

.1336 

. 1787 

. 8213 

54 

36 

28 

7 

. 7127 

. 2873 

.1219 

. 3432 

.8715 

.1338 

. 1801 

. 8199 

53 

32 

32 

8 

. 7152 

. 2847 

.1208 

. 3470 

.8702 

.1340 

. 1815 

. 8185 

52 

28 

36 

9 

. 7178 

. 2822 

.1196 

. 3507 

.8689 

.1341 

. 1828 

. 8171 

51 

24 

40 

10 

.47204 

.52796 

2.1185 

.53545 

1.8676 

1.1343 

.11812 

.88158 

50 

20 

44 

11 

. 7229 

. 2770 

.1173 

. 3582 

.8663 

.1345 

. 1856 

. 8114 

49 

16 

48 

12 

. 7255 

. 2745 

.1162 

. 3619 

.8650 

.1347 

!. 1870 

. 8130 

48 

12 

.62 

13 

. 7281 

. 2719 

.1150 

. 3657 

.8637 

.1349 

. 1883 

. 8117 

47 

8 

56 

14 

. 7306 

. 2694 

.1139 

. 3694 

.86 24 

.1350 

. 1897 

. 8103 

46 

4 

53 

15 

.47332 

.52668 

2.1127 

.53732 

1.8611 

1.1352 

.11911 

.88089 

45 

7 

4 

16 

. 7357 

. 2642 

.1116 

. 3769 

.8598 

.1354 

. 1925 

. 8075 

44 

56 

8 

17 

. 7383 

. 2617 

.1101 

. 3807 

.3585 

.1356 

. 1938 

. 8001 

43 

52 

12 

18 

. 7409 

. 2591 

.1093 

. 3841 

.8572 

.1357 

1952 

. 804S 

42 

48 

16 

19 

. 7434 

. 2565 

.10S2 

. 3882 

.8559 

.1359 

. 1966 

. 8034 

41 

41 

20 

20 

.47460 

.52510 

2.1070 

.53919 

1.8546 

1.1361 

.11980 

.88020 

40 

40 

24 

21 

. 7486 

. 2514 

.1059 

. 3957 

.8533 

.1363 

. 1994 

. 80 !6 

39 

36 

28 

22 

. 7511 

. 2489 

.1048 

. 3995 

.8520 

.1365 

. 2007 

. 7992 

38 

32 

32 

23 

. 7537 

. 2463 

.1036 

. 4032 

.8507 

.1366 

. 2021 

. 7979 

37 

28 

36 

24 

. 7562 

. 2137 

.1025 

. 4070 

.8445 

.1368 

.2035 

. 7965 

36 

24 

40 

25 

.47588 

.52112 

2.1014 

.54107 

1.8482 

1.1370 

.12049 

.87951 

35 

20 

41 

26 

. 7613 

. 2386 

.1002 

. 4145 

.8469 

.1372 

. 2063 

. 7937 

34 

16 

48 

27 

. 7639 

. 2361 

.0991 

. 4183 

.8456 

.1373 

. 2077 

. 7923 

33 

12 

52 

28 

. 7665 

. 2335 

.0980 

. 4220 

.8 443 

.1375 

. 2090 

. 7909 

o2 

8 

56 

29 

. 7690 

. 2310 

.0969 

. 4258 

.8430 

.1377 

. 2104 

. 7: 95 

31 

4 

54: 

30 

.47716 

.52281 

2.0957 

.54295 

1.8418 

1.1379 

.12118 

.87882 

30 

6 

4 

31 

. 7711 

. 2258 

.0946 

. 4333 

.8405 

.1381 

. 2132 

. 7868 

29 

56 

8 

32 

. 7767 

. 2233 

.0935 

. 4371 

.8342 

.1382 

. 2146 

. 7854 

28 

52 

12 

33 

. 7792 

. 2207 

.0921 

. 4409 

.8379 

.1384 

. 2160 

. 7s40 

27 

48 

16 

34 

. 7818 

. 2182 

.0912 

. 4446 

.8367 

.1386 

. 2174 

. 7826 

26 

41 

20 

35 

.47841 

.52156 

2.0901 

.5 4484 

1.8354 

1.1388 

.12188 

.87812 

25 

40 

24 

36 

. 7869 

. 2131 

.0890 

. 4522 

.8341 

.1390 

. 2202 

. 7748 

24 

36 

28 

37 

. 7895 

. 2105 

.0879 

. 4559 

.8329 

.1391 

221 C 

. 7784 

23 

32 

32 

38 

. 7920 

. 2080 

.0868 

. 4597 

.8316 

.1393 

. 2229 

. 7770 

22 

-8 

36 

39 

. 7946 

. 2054 

.0857 

. 4635 

.8303 

.1395 

. 2243 

. 7756 

21 

24 

40 

40 

.47971 

.52029 

2.0846 

.54673 

1.8291 

1.1397 

.12257 

.87742 

20 

20 

44 

41 

. 7997 

. 2003 

.0835 

. 4711 

.8278 

.1399 

. 2271 

. 7728 

19 

16 

48 

42 

. 8022 

. 1978 

.0824 

. 4-74S 

.8265 

.1401 

. 2285 

. 7715 

18 

12 

52 

43 

. 8048 

. 1952 

.0S12 

. 4786 

.8253 

.1402 

. 2299 

. 7701 

17 

8 

56 

41 

. 8073 

. 1927 

.0801 

. 4824 

.8210 

.14/4 

. '.313 

. 70 7 

16 

4 

55 

45 

.48099 

.51901 

2.07 90 

.54862 

1.8227 

1.1406 

.12327 

.87673 

15 

5 

4 

46 

. 8124 

. 1876 

.0779 

. 4900 

.8215 

.1408 

. 2341 

. 7659 

14 

56 

8 

47 

. 8150 

. 1850 

.0768 

. 4937 

.8202 

.1410 

. 2355 

. 7645 

13 

52 

12 

48 

. 8175 

. 1825 

.0757 

. 4975 

.8190 

.1411 

. 2369 

. 7631 

12 

48 

16 

49 

. 8201 

. 1799 

.0746 

. 5013 

.3177 

.1413 

. 2383 

. 7617 

11 

44 

20 

59 

.48226 

.51774 

2.0735 

.55051 

1.8165 

1.1415 

.12397 

.87003 

10 

40 

24 

51 

. 8252 

. 1748 

.0725 

. 5089 

.8152 

.1417 1 

. 2411 

. 7588 

9 

36 

28 

52 

. 8277 

. 1723 

.0714 

. 5127 

.8140 

.1419 

. 2425 

. 7574 

8 

32 

32 

53 

. 8303 

. 1697 

.0703 

. 5165 

.3127 

.1421 

. 2439 

. 7560 

7 

28 

36 

54 

. 8328 

. 1672 

.0692 

. 5203 

.8115 

.1422 

. 2153 

, 7546 

6 

24 

40 

55 

.48351 

.51646 

2 0681 

.55241 

1.8102 

1.1421 

.1246S 

.87-'32 

5 

20 

44 

56 

. 8379 

. 1621 

.0670 

. 5279 

.8090 

.1426 

. 2482 

. 7518 

4 

16 

48 

57 

. 8405 

. 1595 

.0659 

. 5317 

.8078 

.1428 

. 2496 

. 7501 

3 

12 

52 

58 

. 8430 

. 1570 

.6618 

. 5355 

.8065 

.1430 

. 2510 

. 7490 

2 

8 

56 

59 

. 84'>5 

. 1541 

.0637 

. 5393 

.8053 

.1432 

. 2524 

. 7476 

1 

4 

5 <> 

60 

. 8481 

. 1519 

.0627 

. 5431 

.8040 

.1433 1 

. 2538 

. 7462 

0 

4 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secante. 

Cotang. 

Tangent. 

Cosec'nt Vrs. Cos 

Sine. 

M 

M S. 

7 h 

118* 

D 



Natural. 

• 




61° 

4 h 







































Natural Lines, 


2M 


l u 

29° 

Natural Trigonometrical Functions 

150° 

10 h 

M. S. 

M 

Sine. 

Vrs. Cos. 

Cosec'nte 

Tang. 

Co tang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

50 

0 

.48)81 

.51519 

2.0627 

.55431 

1.8040 

1.1433 

.12538 

.87462 

60 

4: 

4 

1 

. 8506 

. 14y3 

.0616 

. 5169 

.8028 

.1435 

. 2552 

. 7448 

59 

56 

8 

2 

. 8532 

. 1468 

.0605 

. 5507 

.8016 

.1437 

. 2566 

. 7434 

58 

52 

12 

3 

. 8557 

. 1443 

.0594 

. 5545 

.8003 

.1439 

. 2580 

. 7420 

57 

43 

16 

4 

. 8583 

. 1417 

.0583 

. 5583 

.7991 

.1441 

. 2594 

. 7405 

56 

44 

20 

5 

.48608 

.51392 

2.0573 

.55621 

1.7979 

1.1443 

.12609 

.87391 

55 

40 

24 

6 

. 8633 

. 1366 

.0562 

. 5659 

.7966 

.1445 

. 2623 

. 7377 

54 

36 

28 

7 

. 8669 

. 1341 

.0551 

. 5697 

.7951 

.1446 

. 2637 

. 7)563 

53 

32 

32 

8 

. 8684 

. 1316 

.0540 

. 5735 

.7942 

.1448 

. 2651 

. 7349 

52 

28 

36 

c 

. 8710 

. 1290 

.0530 

. 5774 

.7930 

.1450 

. 2665 

. 7335 

51 

24 

40 

10 

.48735 

.51265 

2.0519 

.55812 

1.7917 

1.1452 

.12679 

.87320 

50 

20 

44 

n 

. 8760 

. 1239 

.0508 

. 5850 

.7905 

.1454 

. 2694 

. 7306 

49 

16 

48 

12 

. 8786 

. 1214 

.0498 

. 5888 

.7893 

.1456 

. 2708 

. 7292 

48 

12 

62 

13 

. 8811 

. 1189 

.0487 

. 5926 

.7881 

.1158 

. 2722 

. 727S 

47 

8 

66 

14 

. 8837 

. 1163 

.0476 

. 5964 

.7868 

.1459 

. 2736 

. 7264 

46 

4 

57 

15 

.48862 

.51138 

2.0166 

.56003 

1.7856 

1.1461 

.12750 

.87250 

45 

3 

4 

16 

. 8887 

. 1112 

.0455 

. 6041 

.7844 

.1463 

. 2765 

. 7235 

44 

56 

8 

17 

. 8913 

. 1087 

.0414 

. 6079 

.7832 

.1465 

. 2779 

. 7221 

43 

52 

12 

18 

. 8938 

. 1062 

.0134 

. 6117 

.7820 

.1467 

. 2793 

. 7207 

42 

4S 

16 

19 

. 8964 

. 1036 

.0423 

. 6156 

.7808 

.1469 

. 2807 

. 7193 

41 

41 

20 

20 

.48989 

.51011 

2.0113 

.56194 

1.7795 

1.1471 

.12821 

.87178 

40 

40 

24 

21 

. 9014 

. 0986 

.0402 

. 6232 

.7783 

.1473 

. 2836 

. 7164 

39 

36 

28 

22 

. 9040 

. 0960 

.0392 

. 6270 

.7771 

.1474 

. 2850 

. 7150 

38 

32 

32 

23 

. 9065 

. 0935 

.0381 

. 6309 

.7759 

.1476 

. 2864 

. 7130 

37 

28 

36 

24 

. 9090 

. 0910 

.0370 

. 6347 

.7747 

.1478 

. 2879 

. 7121 

36 

24 

40 

25 

.49116 

.50884 

2.0360 

.56385 

1.7735 

1.1480 

.12893 

.87107 

85 

20 

44 

26 

. 9141 

. 0859 

.0349 

. 6424 

.7723 

.1482 

. 2907 

. 7093 

34 

16 

48 

27 

. 9166 

. 0834 

.0339 

. 6462 

.7711 

.1484 

. 2921 

. 7078 

33 

12 

52 

28 

. 9192 

. 0808 

.0329 

. 6500 

.7699 

.1486 

. 2936 

. 7004 

32 

8 

56 

29 

. 8217 

. 0783 

.0318 

. 6539 

.7687 

.1488 

. 2950 

. 7050 

31 

4 

58 

30 

.49242 

.5075S 

2.0308 

.56577 

1.7675 

1.1489 

.12964 

.87035 

30 

2 

4 

31 

. 9268 

. 0732 

•0297 

. 6616 

.7663 

.1491 

. 2979 

. 7021 

29 

56 

8 

32 

. 9293 

. 0707 

•0287 

. 6654 

.7651 

.1493 

. 2993 

. 7007 

28 

52 

12 

33 

. 9318 

. 0682 

■0276 

. 6692 

.7639 

.1495 

. 3fc)7 

. 0992 

27 

48 

16 

34 

. 9343 

. 0656 

.0266 

. 6731 

.7627 

.1497 

. 3022 

. 6978 

26 

41 

20 

35 

.49369 

.50631 

2.0256 

.56769 

1.7615 

1.1499 

.13036 

.86904 

25 

40 

24 

36 

. 9394 

. 0606 

•0245 

. 6808 

.7 603 

.1501 

. 3050 

. 6949 

24 

36 

28 

37 

. 9419 

. 0580 

.0235 

. 6846 

.7591 

.1503 

3065 

. 6935 

23 

32 

32 

38 

. 9445 

. 0555 

.0224 

. 6885 

.7579 

.1505 

. 3079 

. 6921 

22 

28 

36 

39 

. 9470 

. 0530 

.0211 

. 6923 

.7567 

.1507 

. 3094 

. 6906 

21 

24 

40 

40 

.49495 

.50505 

2.0204 

.56962 

1.7555 

1.1508 

.13108 

.86892 

20 

20 

44 

41 

. 9521 

. 0479 

.0194 

. 7000 

.7514 

.1510 

. 3122 

. 6877 

19 

16 

48 

42 

. 9516 

. 0454 

.0183 

. 7039 

.7532 

.1512 

. 3137 

. 6863 

IS 

12 

52 

43 

. 8571 

. 0429 

.0173 

. 7u77 

.7520 

.1514 

. 3151 

. 6849 

17 

8 

56 

41 

. 9596 

. 0404 

.0163 

. 7116 

.7508 

.1516 

. 3166 

. 6834 

16 

4 

5‘J 

45 

.49622 

.50378 

2.0152 

.57155 

1.7496 

1.1518 

.13180 

.86820 

15 

1 

4 

if 

. 9647 

. 0353 

.0142 

. 7193 

.7484 

.1520 

. 3194 

. 6805 

14 

56 

.8 

47 

. 9672 

. 0328 

.0132 

. 7232 

.7473 

.1522 

. 3209 

. 6791 

13 

52 

12 

48 

. 9697 

. 0303 

.0122 

. 7270 

.7461 

.1524 

. 3223 

. 6776 

12 

48 

16 

49 

. 9723 

. 0277 

.0111 

. 7309 

.7449 

.1526 

. 3238 

. 6762 

11 

44 

20 

50 

.49718 

.50252 

2.0101 

.57348 

1.7437 

1.1528 

.13252 

.86748 

10 

40 

24 

51 

. 9773 

. 0227 

.0091 

. 7386 

.7426 

.1530 

. 3267 

. 6733 

9 

36 

28 

52 

. 9798 

. 0202 

.0081 

. 7425 

.7414 

.1531 

. 3281 

. 6719 

8 

32 

32 

53 

. 9823 

. 0176 

.0071 

. 7464 

.7402 

.1533 

. 3296 

. 6704 

7 

2S 

36 

54 

. 9849 

. 0151 

.0061 

. 7502 

.7390 

.1535 

. 3310 

. 6690 

6 

24 

40 

55 

.49874 

.50126 

2 0050 

.57541 

1.7379 

1.1537 

.13325 

.86075 

5 

20 

44 

56 

. 9899 

. 0101 

.0040 

. 7580 

.7367 

.1539 

. 3339 

. 6661 

4 

16 

48 

57 

. 9924 

. 0076 

.0030 

. 7619 

.7355 

.1541 

. 3354 

. 6646 

3 

12 

62 

58 

. 9950 

. 0050 

.0020 

. 7657 

.7344 

.1543 

. 3368' 

. 6632 

2 

8 

56 

59 

. 9975 

. 0025 

.0010 

. 7696 

.7332 

.1545 

. 3383 

. 6617 

1 

4 

GO 

60 

.50000 

. 0000 

•oooo 

. 7735 

.7320 

.1547 

. 3397 

. 6602 

0 

O 

M.S. 

M 

Coslue. 

Vrs.Sin. 

Secante. 

Cotang. 

Tangent. 

Cosec’nt 

Vrs. Cos 

Sine. 

M 

M S. 

?h 

119° 



Natural.. 




60° 

4 h 





























Natural Likes. 239 


2 h 

30° 

Natural Trigonometrical Functions. 

149° 

1 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

JCosec’nte 

Tang. 

Cotang. 

Secante 

| Vrs.Sin 

Cosine. 

1 M 

M.S. 

0 

0 

.50000 

.50000 

2.9000 

.57735 

1.7320 

1.1547 

.13397 

.86602 

60 

«<) 

4 

1 

. 0025 

.49975 

1.9990 

. 7774 

.7309 

.1549 

. 3412 

. 6588 

59 

56 

8 

2 

. 0050 

. 9950 

.9980 

. 7813 

.7297 

.1551 

. 3426 

. 6573 

58 

52 

12 

3 

. 0075 

. 9924 

.9970 

. 7851 

.7286 

.1553 

. 3441 

. 6559 

57 

48 

16 

4 

. 0101 

. 9899 

.9960 

. 7890 

.7274 

.1555 

. 3456 

. 6544 

56 

44 

20 

5 

.50126 

.49874 

1.9950 

.57929 

1.7262 

1.1557 

.13470 

.86530 

55 

40 

1 24 

6 

. 0151 

. 9849 

.9940 

. 7968 

.7251 

.1559 

. 3485 

. 6515 

54 

36 

28 

7 

. 0176 

. 9824 

.9930 

. 8007 

.7239 

.1561 

. 3499 

. 6500 

53 

32 

.12 

8 

. 0201 

. 9799 

.9920 

. 8046 

.7228 

.1562 

. 3514 

. 6486 

52 

28 

36 

9 

. 0226 

. 9773 

.9910 

. 8085 

.7216 

.1564 

. 3529 

. 6471 

51 

24 

40 

10 

.50252 

.49748 

1.9900 

.58123 

1.7205 

1.1566 

.13543 

.86457 

50 

20 

44 

11 

. 0277 

. 9723 

.9890 

. 8162 

.7193 

.1568 

. 3558 

. 6442 

49 

16 

48 

12 

. 0302 

. 9698 

.9880 

. 8201 

.7182 

.1570 

. 3572 

. 6427 

48 

12 

52 

13 

. 0327 

. 9673 

.9870 

. 8240 

.7170 

.1572 

. 3587 

. 6413 

47 

8 

56 

14 

. 0352 

. 9648 

.9860 

. 8279 

.7159 

.1574 

. 3602 

. 6398 

46 

4 

1 

15 

.50377 

.49623 

1.9850 

.58318 

1.7147 

1.1576 

.13616 

.86383 

45 

59 

4 

16 

. 0402 

. 9597 

.9840 

. 8357 

.7136 

.1578 

. 3631 

. 6369 

44 

56 

8 

17 

. 0428 

. 9572 

.9830 

. 8396 

.7124 

.1580 

. 3646 

. 6354 

43 

52 

12 

18 

. 0453 

. 9547 

.9820 

. 8435 

.7113 

.1582 

. 3660 

. 6339 

42 

48 

16 

19 

. 0478 

. 9522 

.9811 

. 8474 

.7101 

.1584 

. 3675 

. 6325 

41 

44 

20 

20 

.50503 

.49497 

1.9801 

.58513 

1.7090 

1.1586 

.13690 

86310 

40 

40 

24 

21 

. 0528 

. 9472 

.9791 

. 8552 

.7079 

.1588 

. 3704 

. 6295 

39 

36 

28 

22 

. 0553 

. 9447 

.9781 

. 8591 

.7067 

.1590 

. 3719 

. 6281 

38 

32 

32 

23 

. 0578 

. 9422 

.9771 

. 8630 

.7056 

.1592 

. 3734 

. 6266 

37 

28 

36 

24 

. 0603 

. 9397 

.9761 

. 8670 

.7044 

.1594 

. 3749 

. 6251 

36 

24 

40 

25 

.50628 

.49371 

1.9752 

.58709 

1.7033 

1.1596 

.13763 

.86237 

35 

20 

44 

26 

. 0653 

. 9346 

.9742 

. 8748 

.7022 

.1598 

. 3778 

. 6222 

34 

16 

48 

27 

. 0679 

. 9321 

.9732 

. 8787 

.7010 

.1600 

. 3793 

. 6207 

33 

12 

52 

28 

. 0704 

. 9296 

.9722 

. 8826 

.6999 

.1602 

. 3807 

. 6192 

32 

8 

56 

29 

. 0729 

. 9271 

.9713 

. 8865 

.6988 

.1604 

. 3822 

. 6178 

31 

4 

2 

30 

.50754 

.49246 

1.9703 

.58904 

1.6977 

1.1606 

.13837 

.86163 

30 

58 

4 

31 

. 0779 

. 9221 

.9693 

. 8944 

.6965 

.1608 

. 3852 

. 6148 

29 

56 

8 

32 

. 0804 

. 9196 

.9683 

. 8983 

.6954 

.1610 

. 3867 

. 6133 

28 

52 

12 

33 

. 0829 

. 9171 

.9674 

. 9022 

.6943 

.1612 

. 3881 

. 6118 

27 

48 

16 

34 

. 0854 

. 9146 

.9664 

. 9061 

.6931 

.1614 

. 3896 

. 6104 

26 

44 

20 

35 

.50879 

.49121 

1.9654 

.59100 

1.6920 

1.1616 

.13911 

.86089 

25 

40 

24 

36 

. 0904 

. 9096 

.9645 

. 9140 

.6909 

.1618 

. 3926 

.,6074 

24 

36 

28 

37 

. 0929 

. 9071 

.9635 

. 9179 

.6898 

.1620 

. 3941 

. 6059 

23 

32 

32 

38 

. 0954 

. 9046 

.9625 

. 9218 

.6887 

.1622 

. 3955 

. 6044 

22 

28 

36 

39 

. 0979 

. 9021 

.9616 

. 9258 

.6875 

.1624 

. 3970 

. 6030 

21 

24 

40 

40 

.51004 

.48996 

1.9606 

.59297 

1.6864 

1.1626 

.13985 

.86015 

20 

20 

44 

41 

. 1029 

. 8971 

.9596 

. 9336 

.6S53 

.1628 

. 4000 

. 6000 

19 

16 

48 

42 

. 1054 

. 8946 

.9587 

. 9376 

.6842 

.1630 

. 4015 

. 5985 

18 

12 

52 

43 

. 1079 

. 8921 

.9577 

. 9415 

.6831 

.1632 

. 4030 

. 5970 

17 

8 

56 

44 

. 1104 

. 8896 

.9568 

. 9454 

.6820 

.1634 

. 4044. 

. 5955 

16 

4 

3 

45 

.51129 

.48871 

1.9558 

.59494 

1.6808 

1.1636 

.14059 

.85941 

15 

57 

4 

46 

. 1154 

. 8846 

.9549 

. 9533 

.6797 

.1638 

. 4074 

. 5926 

14 

56 

8 

47 

. 1179 

. 8821 

.9539 

. 9572 

.6786 

.1640 

. 4089 

. 5911 

13 

52 

12 

48 

. 1204 

. 8796 

.9530 

. 9612 

.6775 

.1642 

. 4104 

. 5896 

12 

48 

16 

49 

. 1229 

. 8771 

.9520 

. 9651 

*.6764 

.1644 

. 4119 

. 5881 

11 

44 

20 

50 

.51254 

.48746 

1.9510 

.59691 

1.6753 

1.1646 

.14134 

.85866 

10 

40 

24 

51 

. 1279 

. 8721 

.9501 

. 9730 

.6742 

.1648 

. 4149 

. 5851 

9 

36 

28 

52 

. 1304 

. 8696 

.9491 

. 9770 

.6731 

.1650 

. 4164 

. 5836 

8 

32 

32 

53 

. 1329 

. 8671 

.9482 

. 9809 

.6720 

.1652 

. 4178 

. 5821 

7 

28 

36 

54 

. 1354 

. 8646 

.9473 

. 9849 

.6709 

.1654 

. 4193 

. 5806 

6 

24 

40 

55 

.51379 

.18621 

1.9463 

.59888 

1.6698 

1.1656 

.14208 

.85791 

5 

20 

44 

56 

. 1404 

. 8596 

.9454 

. 9928 

.6687 

.1658 

. 4223 

. 5777 

4 

16 

48 

57 

. 1429 

. 8571 

.9444 

. 9967 

.6676 

.1660 

. 4238 

. 5762 

3 

12 

52 

58 

. 1454 

. 8546 

.9435 

.60007 

.6665 

.1662 

. 4253 

. 5747 

2 

8 

56 

59 

. 1479 

. 8521 

.9425 

. 0046 

.6654 

.1664 

. 4268 

. 5732 

1 

4 

4 

60 

. 1504 

. 8496 

.9416 

. 0086 

.6643 

.1666 

. 4283 

. 5717 

0 

56 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secante. 

CotaugJ 

Tangent. 

Cosec'nti 

Vrs.Cos 

Sine. 

M 

M.S. 

8 h 

120° 



Natural. 



59° 

3 h 













































240 Natural Lines. 


2 h 

31° 

Natural Trigonometrical Functions 

148° 

9 h 

M.S. 

M 

Sine. 

Vrs. Cos. 

iCosec’nte 

Tang. 

Cotang. 

Seeante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

4 

0 

.51504 

.48496 

1.9416 

.60086 

1.6643 

1.1666 

.14283 

.85717 

60 

56 

4 

1 

. 1529 

. 8471 

.9407 

. 0126 

.6632 

.1668 

. 4298 

. 5702 

59 

56 

8 

2 

. 1554 

. 8446 

.9397 

. 0165 

.6621 

.1670 

. 4313 

. 5687 

58 

52 

12 

3 

. 1578 

. 8421 

.9388 

. 0205 

.6610 

.1672 

. 4328 

. 5672 

57 

48 

16 

4 

. 1603 

. 8396 

.9378 

. 0244 

.6599 

.1674 

. 4343 

. 5657 

56 

44 

20 

5 

.51628 

.48371 

1.9369 

.60284 

1.6588 

1.1676 

.14358 

.85642 

55 

40 

24 

6 

. 1653 

. 8347 

.9360 

. 0324 

.6577 

.1678 

. 4373 

. 5627 

54 

36 

28 

7 

. 1678 

. 8322 

.9350 

. 0363 

.6566 

.1681 

. 4388 

. 5612 

53 

32 

32 

8 

. 1703 

. 8297 

.9341 

. 0403 

.6555 

.1683 

. 4403 

. 5597 

52 

28 

36 

9 

. 1728 

. 8272 

.9332 

. 0443 

.6544 

.1685 

. 4418 

. 5582 

51 

24 

: 40 

10 

.51753 

.48247 

1.9322 

.60483 

1.6534 

1.1687 

.14433 

.85566 

50 

20 

44 

11 

. 1778 

. 8222 

.9313 

. 0522 

.6523 

.1689 

. 4448 

. 5551 

49 

16 

^ 48 

12 

. 1803 

. 8197 

.9304 

. 0562 

.6512 

.1691 

. 4463 

. 5536 

48 

12 

62 

13 

. IS 27 

. 8172 

.9295 

. 0602 

.6501 

.1693 

. 4479 

. 5521 

47 

8 

66 

14 

. 1852 

. 8147 

.9285 

. 0642 

.6490 

.1695 

. 4494 

. 5506 

46 

4 

5 

15 

.51S77 

.48123 

1.9276 

.60681 

1.6479 

1.1697 

.14509 

.85491 

45 

55 

4 

16 

. 1902 

. 8098 

.9267 

. 0721 

.6469 

.1699 

. 4524 

. 5476 

44 

56 

8 

17 

. 1927 

. 8073 

.9258 

. 0761 

.6458 

.1701 

. 4539 

. 5461 

43 

52 

12 

18 

. 1952 

. 8048 

.9248 

. 0801 

.6447 

.1703 

. 4554 

. 5446 

42 

48 

16 

19 

. 1977 

8023 

.9239 

. 0841 

.6436 

.1705 

. 4569 

. 6431 

41 

44 

20 

20 

.52002 

.47998 

1.9230 

.60881 

1.6425 

1.1707 

.14584 

.85416 

40 

40 

24 

21 

. 2026 

. 7973 

.9221 

. 0920 

.6415 

.1709 

. 4599 

. 5400 

39 

36 

28 

22 

. 2051 

. 7949 

.9212 

. 0960 

.6404 

.1712 

. 4615 

. 5385 

38 

32 

32 

23 

. 2076 

. 7924 

.9203 

. 1000 

.6393 

.1714 

. 4630 

. 5370 

37 

28 

36 

24 

. 2101 

. 7899 

.9193 

. 1040 

.6383 

.1716 

. 4645 

. 5355 

36 

24 

40 

25 

.52126 

.47874 

1.9184 

.61080 

1.6372 

1.1718 

.14660 

.85340 

35 

20 

44 

26 

. 2151 

. 7849 

.9175 

. 1120 

.6361 

.1720 

. 4675 

. 5325 

34 

16 

i 48 

27 

. 2175 

. 7824 

.9166 

. 1160 

.6350 

.1722 

. 4690 

. 5309 

33 

12 

52 

28 

. 2200 

. 7800 

.9157 

. 1200 

.6340 

.1724 

. 4706 

. 5294 

32 

8 

56 

29 

. 2225 

. 7775 

.9148 

. 1240 

.6329 

.1726 

. 4721 

. 5279 

31 

4 

6 

30 

.52250 

.47750 

1.9139 

.61280 

1.6318 

1.1728 

.14736 

.85264 

30 

54 

4 

31 

. 2275 

. 7725 

.9130 

. 1320 

.6308 

.1730 

. 4751 

. 5249 

29 

56 

8 

32 

. 2299 

. 7700 

.9121 

. 1360 

.6297 

.1732 

. 4766 

. 5234 

28 

52 

12 

33 

. 2324 

. 7676 

.9112 

. 1400 

.6286 

.1734 

. 4782 

. 5218 

27 

48 

16 

34 

. 2319 

. 7651 

.9102 

. 1440 

.6276 

.1737 

. 4797 

. 5203 

26 

44 

20 

35 

.52374 

.47626 

1.9093 

.61480 

1.6265 

1.1739 

.14812 

.85188 

25 

40 

24 

36 

. 2398 

. 7601 

.9084 

. 1520 

.6255 

.1741 

. 4827 

5173 

24 

36 

28 

37 

. 2423 

. 7577 

.9075 

. 1560 

.6244 

.1743 

. 4842 

. 5157 

23 

32 

32 

38 

. 2448 

. 7552 

.9066 

. 1601 

.6233 

.1745 

. 4858 

. 5142 

22 

28 

36 

39 

. 2473 

. 7527 

.9057 

. 1641 

.6223 

.1747 

. 4873 

. 5127 

21 

24 

40 

40 

.52498 

.47502 

1.9048 

.61681 

1.6212 

1.1749 

.14888 

.85112 

20 

20 

44 

41 

. 2522 

. 7477 

.9039 

. 1721 

.6202 

.1751 

. 4904 

. 5096 

19 

16 

48 

42 

. 2547 

. 7453 

.9030 

. 1761 

.6191 

.1753 

. 4919 

. 5081 

IS 

12 

52 

43 

. 2572 

. 7428 

.9021 

. 1801 

.6181 

.1756 

. 4934 

. 5066 

17 

8 

56 

44 

. 2597 

. 7403 

.9013 

. 1842 

.6170 

.1758 

. 4949 

. 5050 

16 

4 

7 

45 

.52621 

.47379 

1.9004 

.61882 

1.6160 

1.1760 

.14965 

.85035 

15 

53 

4 

46 

. 2646 

. 7354 

.8995 

. 1922 

.6149 

.1762 

. 4980 

. 5020 

14 

56 

8 

47 

. 2671 

. 7329 

.8986 

. 1962 

.6139 

.1764 

. 4995 

. 5004 

13 

52 

12 

48 

. 2695 

. 7304 

.8977 

. 2003 

.6128 

.1766 

. 5011 

. 4989 

12 

48 

16 

49 

. 2720 

. 7280 

.8968 

2«43 

.6118 

.1768 

. 5026 

. 4974 

11 

44 

20 

50 

.52745 

.47255 

1.8959 

.62083 

1.6107 

1.1770 

.15041 

.84959 

10 

40 

24 

51 

. 2770 

. 7230 

.8950 

. 2123 

.6097 

.1772 

. 5057 

. 4943 

9 

36 

28 

52 

. 2794 

. 7205 

.8941 

. 2164 

.6086 

.1775 

. 5072 

. 4928 

8 

32 

32 

53 

. 2819 

. 7181 

.8932 

. 2204 

.6076 

.1777 

. 5087 

. 4912 

7 

28 

36 

54 

. 2844 

. 7166 

.8924 

. 2244 

.6066 

.1779 

. 5103 

. 4897 

6 

24 

40 

55 

.5286S 

.47131 

1.8915 

.62285 

1.6055 

1.1781 

.15118 

.84882 

5 

20 

44 

56 

. 2893 

. 7107 

.8906 

. 2325 

.6045 

.1783 

. 5133 

. 4866 

4 

16 

48 

57 

. 2918 

. 7082 

.8897 

. 2366 

.6034 

.1785 

. 5149 

. 4851 

3 

12 

52 

58 

. 2942 

. 7057 

.8888 

. 2406 

.6024 

.1787 

. 5164 

. 4836 

2 

8 

56 

59 

. 2967 

. 7033 

.8879 

. 2446 

.6014 

.1790 

. 5180 

. 4820 

1 

4 

8 

60 

. 2992 

. 7008 

.8871 

. 24S7 

.6003 

.1792 

. 5195 

. 4805 

0 

52 

M. S. 

M 

Cosine. 

Vrs. Sin. 

Seeante. 

Cotang.[ 

Tangent. 

Cosee’ut 

Vrs. Cos 

Sine. 

M 

M.S. 

gu 

121° 



Natural. 




58° 

3 h 





































Natural Lines. 


241 


2 h 

32° Natural Trigonometrical Functions. 

* 147° 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

"Cosec’n te 

Tang. 

Cotang. 

Secante 

JVrs.Sin 

Cosine. 

M 

M.S. 

8 

0 

.52992 

.47008 

1.8871 

.62487 

1.6003 

1.1792 

.15195 

.81805 

60 

52 

4 

1 

. 3016 

. 6983 

.8862 

. 2527 

.5993 

.1791 

. 5211 

. 4789 

59 

56 

8 

2 

. 3041 

. 6959 

.8853 

. 2568 

.5983 

.1796 

. 5226 

. 4774 

58 

52 

12 

3 

. 3o66 

. 6934 

.8844 

. 2608 

.5972 

.1798 

. 5211 

. 4758 

57 

48 

16 

4 

. 3090 

. 6909 

.8836 

. 2649 

.5962 

.1800 

. 5257 

. 4743 

56 

44 

20 

5 

.53115 

.46885 

1.8827 

.62689 

1.5952 

1.1802 

.15272 

.84728 

55 

40 

24 

6 

. 3140 

. 6860 

.8818 

. 2730 

.5941 

.1805 

. 5288 

. 4712 

54 

36 

28 

7 

. 3164 

. 6835 

.8809 

. 2770 

.5931 

.1807. 

. 5303 

. 4697 

53 

32 

32 

8 

. 3189 

. 6811 

.8801 

. 2811 

.5921 

.1809 

. 5319 

. 4681 

52 

28 

36 

9 

. 3214 

. 6786 

.8792 

. 2851 

.5910 

.1811 

. 5334 

. 1666 

51 

24 

40 

10 

.53238 

.46762 

1.8783 

.62892 

1.5900 

1.1813 

.15350 

.84650 

50 

20 

44 

11 

. 3263 

. 6737 

.8775 

. 2933 

.5890 

.1815 

. 5365 

. 4635. 

49 

16 

48 

12 

. 3288 

. 6712 

.8766 

. 2973 

.5880 

.1818 

. 5381 

. 4619 

48 

12 

52 

13 

. 3312 

. 6688 

.8757 

. 3014 

.5869 

.1820 

. 5396 

. 4604 

47 

8 

56 

14 

. 3337 

. 6663 

.8749 

. 3055 

.5859 

.1822 

. 5412 

. 4588 

4G 

4 

9 

15 

.53361 

.46638 

1.3740 

.63095 

1.5849 

1.1824 

.15427 

.84573 

45 

51 

4 

16 

. 33S6 

. 6614 

.8731 

. 3136 

.5839 

.1826 

. 5143 

. 4557 

44 

56 

8 

17 

. 3411 

. 6589 

.8723 

. 3177 

.5829 

.1828 

. 5458 

. 4542 

43 

52 

12 

18 

. 3435 

. 6565 

.8714 

. 3217 

.5818 

.1831 

. 5174 

. 4526 

42 

48 

16 

19 

. 3460 

. 6540 

.8706 

. 3258 

.5808 

.1833 

. 5489 

. 4511 

41 

44 

20 

20 

.53484 

.46516 

1.8697 

.63299 

1.5798 

1.1835 

.15505 

.84495 

40 

40 

24 

21 

. 3509 

. 6491 

.8688 

. 3339 

.5788 

.1837 

. 5520 

. 4479 

39 

36 

28 

22 

. 3533 

. 6466 

.8680 

. 3380 

.5778 

.1839 

. 5536 

. 4464 

38 

32 

32 

23 

. 3558 

. 6442 

.8671 

. 3421 

.5768 

.1811 

. 5552 

. 4448 

37 

28 

36 

24 

. 3583 

. 6417 

.8663 

. 3462 

.5757 

.1844 

. 5567 

. 4433 

36 

24 

40 

25 

.53607 

.46393 

1.8654 

.63503 

1.5747 

1.1846 

.15583 

.84417 

35 

20 

44 

26 

. 3632 

. 6368 

.8646 

. 3543 

.5737 

.1848 

. 5598 

. 4402 

34 

16 

48 

27 

. 3656 

. 6344 

.8637 

. 3584 

.5727 

.1850 

. 5614 

. 4386 

33 

12 

52 

28 

. 3681 

. 6319 

.8629 

. 3625 

.5717 

.1852 

. 5630. 

. 4370 

32 

8 

56 

29 

. 3705- 

. 6294 

.8620 

. 3666 

.5707 

.1855 

. 5645 

. 4355 

31 

4 

10 

30 

.53730 

.46270 

1.8611 

.63707 

1.5697 

] .1857 

.15661 

.84339 

30 

50 

4 

31 

. 375 4 

. 6245 

.8603 

. 3748 

.5687 

.1859 

. 5676 

. 4323 

29 

56 

8 

32 

. 3779 

. 6221 

.8595 

. 3789 

.5677 

.1861 

. 5692. 

. 4308 

28 

52 

12 

33 

. 3803 

. 6196 

.8586 

. 3830 

.5667 

.1863 

. 5708 

. 4292 

27 

48 

16 

34 

. 3828 

. 6172 

.8578 

. 3871 

.5657 

.1S66 

. 5723 

. 4276 

26 

44 

20 

35 

.53852 

.46147 

1.8569 

.63912 

1.5646 

1.1868 

.15739 

.84261 

25 

40 

24 

36 

. 3877 

. 6123 

.8561 

. 3953 

.5636 

.1870 

. 5755. 

. 4245 

24 

36 

28 

37 

. 3901 

. 6098 

.8552 

. 3994 

.5626 

.1872 

. 5770. 

. 4229. 

23 

32 

32 

38 

. 3926 

. 6074 

.8544 

. 4035 

.5616 

.1874 

. 5786 

. 4214 

22 

28 

36 

39 

. 3950 

. 6049 

.8535 

. 4076 

.5606 

.1877 

. 5802. 

. 4198 

21 

24 

40 

40 

.53975 

.46025 

1.8527 

.64117 

1.5596 

1.1879 

.15817 

.84182 

20 

20 

44 

41 

. 3999 

. 6000 

.8519 

. 4158 

.5586 

.1881 

. 5833 

. 4167 

19 

16 

48 

42 

. 4024 

. 5976 

.8510 

. 4199 

■5577 

.1883 

. 5849 

. 4151 

18 

12 

52 

43 

. 4048 

. 5951 

.8502 

. 4240 

.5567 

.1886 

. 5865 

. 4135 

17 

8 

56 

44 

. 4073 

. 5927 

.8493 

. 4281 

.5557 

.1888 

. 5880 

. 4120 

16 

4 

11 

45 

.54097 

.45902 

1.8485 

.64322 

1.5517 

1.1890 

.15896 

.84104 

15 

49 

4 

46 

. 4122 

. 5878 

.8477 

. 4363 

.5537 

.1892 

. 5912 

. 4088 

14 

56 

8 

47 

. 4146 

. 5854 

.8468 

. 4104 

.5527 

.1894 

. 5927 

. 4072 

13 

52 

12 

48 

. 4171 

. 5S29 

.8460 

. 4146 

.5517 

.1897 

. 5913 

. 4057 

12 

48 

16 

49 

. 4195 

. 5805 

.8452 

. 4487 

.5507 

.1899 

. 5959 

. 4041 

11 

44 : 

20 

50 

.54220 

.45780 

1.8443 

.64528 

1.5497 

1.1901 

.15975 

.84025 

10 

40 * 

24 

51 

. 4244 

. 5756 

.8435 

. 4569 

.5487 

.1903 

. 5991 

. 4009 

9 

36 

28 

52 

. 4268 

. 5731 

.8427 

. 4610 

.5477 

.1906 

. 6006 

. 3993 

8 

32 

32 

53 

. 4293 

. 6707 

.8418 

. 4652 

.5467 

.1908 

. 6022 

. 3978 

7 

28 

36 

54 

. 4317 

. 5682 

.8410 

. 4693 

.5458 

.1910 

. 6038 

. 3962 

6 

24 : 

40 

55 

.54342 

.45658 

1.8402 

.64734 

1.5448 

1.1912 

.16054 

.83946 

5 

20 

44 

56 

. 4366 

. 5034 

.8394 

. 4775 

.5438 

.1915 j 

. 6070 

. 3930 

4 

16 

48 

57 

. 4391 

. 5009 

.8385 

. 4817 

.5128 

.1917 

. 6085 

. 3914 

3 

12 

52 

58 

. 4415 

. 5585 

.8377 

. 4858 

.5418 

.1919 

. 0101 

. 3899 

2 

8 ; 

56 

59 

. 4439 | 

. 6560 

.8369 

. 4899 

.5408 

.1921 

. 6117 

. 3883 

1 

4 

13 

60 

. 4464 | 

. 5536 

.8361 

. 4941 

.5399 

.1922 

. 6133 

. 3867 

0 

48 

M.S. 

8 h 

M 

122 c 

Cosine. | 

Vrs.Sin. 

Secante. 

Cotaug. (Tangent. 

Natural. 

Cosec'ut 1 

Vrs.Cos 

Sine. 

: 

M 

57° 

M.S. 

3 h 


16 





































242 


Natural Lines, 


2 h 

33° 

• 

Natural Trigonometrical Functions 

146° 

9 h 

M.S. 

M 

Sine. 

Vrs. Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

13 

0 

.54464 

.45536 

1.8361 

.64911 

1.5399 

1.1924 

.16133 

.83867 

60 

48 

4 

l 

. 4488 

. 5512 

.8352 

. 4982 

.5389 

.1926 

. 6149 

. 3851 

59 

56 

8 

2 

. 4513 

. 5487 

.8344 

. 5023 

.5379 

.1928 

. 6165 

. 3835 

58 

52 

12 

3 

. 4537 

. 5463 

.8336 

. 5065 

.5369 

.1930 

. 6180 

. 3819 

57 

48 

16 

4 

. 4561 

. 5438 

.8328 

. 5106 

.5359 

.1933 

. 6196 

. 3804 

56 

44 

20 

5 

.54586 

.45414 

1.8320 

.65148 

1.5350 

1.1935 

.16212 

.83788 

55 

40 

24 

6 

. 4610 

. 5390 

.8311 

. 5189 

.5340 

.1937 

. 6228 

. 3772 

54 

36 

28 

7 

. 4634 

. 5365 

.8303 

. 5231 

.5330 

.1939 

. 6244 

. 3756 

53 

32 

32 

8 

. 4659 

. 5341 

.8295 

. 5272 

.5320 

.1942 

. 6260 

. 3740 

52 

28 

36 

9 

. 4683 

. 5317 

.8287 

. 5314 

.5311 

.1944 

. 6276 

. 3724 

51 

24 

40 

10 

.54708 

.45292 

1.8279 

.65355 

1.5301 

1.1946 

.16292 

.83708 

50 

20 

44 

11 

. 4732 

. 5268 

.8271 

. 5397 

.5291 

.1948 

. 6308 

. 3692 

49 

16 

48 

12 

. 4756 

. 5244 

.8263 

. 5438 

.5282 

.1951 

. 6323 

. 3676 

48 

12 

52 

13 

. 4781 

. 5219 

.8255 

. 5480 

.5272 

.1953 

. 6339 

. 3660 

47 

8 

56 

14, 

. 4S05 

. 5195 

.8246 

. 5521 

.5262 

.1955 

. 6355 

. 3644 

46 

4 

13 

15 

.54829 

.45171 

1.8238 

.65563 

1.5252 

1.1958 

.16371 

.83629 

45 

4T 

4 

16 

. 4854 

. 5146 

.8230 

. 5604 

.5243 

.1960 

. 6387 

. 3613 

44 

56 

8 

17 

. 4878 

. 5122 

.8222 

. 5646 

.5233 

.1962 

. 6403 

. 3597 

43 

52 

12 

18 

. 4902 

. 5098 

.8214 

. 5688 

.5223 

.1964 

. 6419 

. 3581 

42 

48 

16 

19 

. 4926 

5073 

.8206 

. 5729 

.5214 

.1967 

. 6435 

. 3565 

41 

44 

20 

20 

.54951 

.45049 

1.8198 

.65771 

1.5204 

1.1969 

.16451 

.83549 

40 

40 

24 

21 

. 4975 

. 5025 

.8190 

. 5813 

.5195 

.1971 

. 6467 

. 3533 

39 

36 

28 

22 

. 4999 

. 5000 

.8182 

. 5854 

.5185 

.1974 

. 6483 

. 3517 

38 

32 

32 

23 

. 5024 

. 4976 

.8174 

. 5896 

.5175 

.1976 

. 6499 

. 3501 

37 

28 

36 

24 

. 5048 

. 4952 

.8166 

. 5938 

.5166 

.1978 

. 6515 

. 3485 

36 

24 

40 

25 

.55072 

.44928 

1.8158 

.65980 

1.5156 

1.1980 

.16531 

.83469 

35 

20 

44 

26 

. 5097 

. 4903 

.8150 

. 6021 

.5147 

.1983 

. 6547 

. 3453 

34 

16 

48 

27 

. 5121 

. 4879 

.8142 

. 6063 

.5137 

.1985 

. 6563 

. 3437 

33 

12 

52 

28 

. 5145 

. 4855 

.8134 

. 6105 

.5127 

.1987 

. 6579 

. 3421 

32 

8 

56 

29 

. 5169 

. 4830 

.8126 

. 6147 

.5118 

.1990 

. 6595 

. 3405 

31 

4 

14 

30 

.55194 

.44806 

1.8118 

.66188 

1.5108 

1.1992 

.16611 

.83388 

30 

4G 

4 

31 

. 5218 

. 4782 

.8110 

. 6230 

.5099 

.1994 

. 6627 

. 3372 

29 

56 

8 

32 

. 5242 

. 4758 

.8102 

. 6272 

.5089 

.1997 

. 6643 

. 3356 

28 

52 

12 

33 

. 5266 

. 4733 

.8094 

. 6314 

.5080 

.1999 

. 6660 

. 3340 

27 

48 

16 

34 

. 5291 

.. 4709 

.8086 

. 6356 

.5070 

.2001 

. 6676 

. 3324 

26 

44 

20 

.35 

.55315 

.44685 

1.8078 

.66398 

1.5061 

1.2004 

.16692 

.83308 

25 

40 

24 

36 

. 5339 

. 4661 

.8070 

. 6440 

.5051 

.2006 

. 6708 

. 3292 

24 

36 

28 

37 

. 5363 

. 4637 

.8062 

. 6482 

.5042 

.2008 

. 6724 

. 3276 

23 

32 

32 

38 

. 5388 

. 4612 

.8054 

. 6524 

.5032 

.2010 

. 6740 

. 3260 

22 

28 

36 

39 

. 5412 

. 4588 

.8047 

. 6566 

.5023 

.2013 

. 6756 

. 3244 

21 

24 

40 

40 

.55436 

.44564 

1.8039 

.66608 

1.5013 

1.2015 

.16772 

.83228 

20 

20 

44 

41 

. 5460 

. 4540 

.8031 

. 6650 

.5004 

.2017 

. 6788 

. 3211 

19 

16 

48 

42 

. 5484 

. 4515 

.8023 

. 6692 

.4994 

.2020 

. 6804 

. 3195 

18 

12 

52 

43 

. 5509 

. 4491 

.8015 

. 6734 

.4985 

.2022 

. 6821 

. 3179 

17 

8 

56 

44 

. 5533 

. 4467 

.8007 

. 6776 

.4975 

.2024 

. 6837 

. 3163 

16 

4 

15 

45 

.55557 

.44443 

1.7999 

.66818 

1.4966 

1.2027 

.16853 

.83147 

15 

45 

4 

46 

. 5581 

. 4419 

.7992 

. 6860 

.4957 

.2029 

. 6869 

. 3131 

14 

56 

8 

47 

. 5605 

. 4395 

.7984 

. 6902 

.4947 

.2031 

. 6885 

. 3115 

13 

52 

12 

48 

. 5629 

. 4370 

.7976 

. 6944 

.4938 

.2034 

. 6901 

. 3098 

12 

48 

16 

49 

. 5654 

. 4346 

.7968 

6986 

.4928 

.2036 

. 6918 

. 3082 

11 

44 

20 

50 

.55678 

.44322 

1.7960 

.67028 

1.4919 

1.2039 

.16934 

.83066 

10 

40 

24 

51 

. 5702 

. 4298 

.7953 

. 7071 

.4910 

.2041 

. 6950 

. 3050 

9 

36 

28 

52 

. 5726 

. 4274 

.7945 

. 7113 

.4900 

.2043 

. 6966 

. 3034 

8 

32 

32 

53 

. 5750 

. 4250 

.7937 

. 7155 

.4891 

.2046 

. 6982 

. 3017 

7 

28 

36 

54 

. 5774 

. 4225 

.7929 

. 7197 

.4881 

.2048 

. 6999 

. 3001 

6 

24 

40 

55 

.55799 

.44201 

1.7921 

.67239 

1.4872 

1.2050 

.17015 

.82985 

5 

20 

44 

56 

. 5823 

. 4177 

.7914 

. 7282 

.4863 

.2053 

. 7031 

. 2969 

4 

16 

48 

57 

. 5847 

. 4153 

.7906 

. 7324 

.4853 

.2055 

. 7047 

. 2952 

3 

12 

52 

58 

. 5871 

. 4129 

.7898 

. 7366 

.4844 

.2057 

. 7064 

. 2936 

2 

8 

56 

59 

. 5895 

. 4.105 

.7891 

. 7408 

.4835 

.2060 

. 7080 

. 2920 

1 

4 

10 

60 

. 5919 

. 4081 

.7883 

. 7451 

.4826 

.2062 

. 7096 

. 2904 

0 

44 

M.S. 

8 h 

M 

123 

Cosine. 

D 

Vrs. Sin. I Secante. | 

Cotaug-ITaugent. 

Natural. 

Cosec'nt I 

Vrs.Cos 

Sine, j 

M 

56° 

M.S. 

3 h 

























Natural Lines. 


243 


2 h 

34° Natural Trigonometrical Functions. 

145° 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

[Cosee’nte 

Tang. 

Cotang. 

Secaute 

jVrs.Sin 

Cosine. 

M 

M.S. 

16 

0 

.55919 

.44081 

1.7883 

.67451 

1.4826 

1.2062 

.17096 

.82904 

60 

44 

4 

1 

. 5943 

. 4057 

.7875 

. 7493 

.4816 

.2064 

. 7112 

. 2887 

59 

56 

8 

2 

. 5967 

. 4032 

.7867 

. 7535 

.4807 

.2067 

. 7129 

. 2871 

58 

52 

12 

3 

. 5992 

. 4008 

.7860 

. 7578 

.4798 

.2069 

. 7145 

. 2855 

57 

48 

16 

4 

. 6016 

. 3984 

.7852 

. 7620 

.4788 

.2072 

. 7161 

. 2839 

56 

44 

20 

5 

.56040 

.43960 

1.7844 

.67663 

1.4779 

1.2074 

.17178 

.82822 

55 

40 

24 

6 

. 6061 

. 3936 

.7837 

. 7705 

.4770 

.2076 

. 7194 

. 2806 

54 

36 

28 

7 

. 6088 

. 3912 

.7829 

. 7747 

.4761 

.2074 

. 7210 

. 2790 

53 

32 

32 

8 

. 6112 

. 3888 

.7821 

. 7790 

.4751 

.2081 

. 7227 

. 2773 

52 

28 

36 

9 

. 6136 

. 3864 

.7814 

. 7832 

.4742 

.2083 

. 7243 

. 2757 

51 

24 

40 

10 

.56160 

.43840 

1.7806 

.67875 

1.4733 

1.2086 

.17259 

.82741 

50 

20 

41 

11 

. 6184 

. 3816 

.7798 

. 7917 

.4724 

.2088 

. 7276 

. 2724 

49 

16 

48 

12 

. 6208 

. 3792 

.7791 

. 7960 

.4714 

.2091 

. 7292 

. 2708 

48 

12 

52 

13 

. 6232 

. 3768 

.7783 

. 8002 

.4705 

.2093 

. 7308 

. 2692 

47 

8 

56 

14 

. 6256 

. 3743 

.7776 

. 8045 

.4696 

.2095 

. 7325 

. 2675 

46 

4 

17 

15 

.56280 

.43719 

1.7768 

.68087 

1.4687 

1.2098 

.17341 

.82659 

45 

43 

4 

16 

. 6304 

. 3695 

.7760 

. 8130 

.4678 

.2100 

. 7857 

. 2643 

44 

56 

8 

17 

. 6328 

. 3671 

.7753 

. 8173 

.4669 

.2103 

. 7374 

. 2626 

43 

52 

12 

18 

. 6353 

. 3647 

.7745 

. 8215 

.4659 

.2105 

.•7390 

. 2610 

42 

48 

16 

19 

. 6377 

. 3623 

.7738 

. 8258 

.4650 

.2107 

. 7406 

. 2593 

41 

44 

20 

20 

.56401 

.43599 

1.7730 

.68301 

1.4041 

1.2110 

.17423 

.82577 

40 

40 

24 

21 

. 6425 

. 3575 

.7723 

. 8343 

.4632 

.2112 

. 7439 

. 2561 

39 

36 

28 

22 

. 6449 

. 3551 

.7715 

. 8386 

.4623 

.2115 

. 7456 

. 2544 

38 

32 

32 

23 

. 6473 

. 3527 

.7708 

. 8429 

.4614 

.2117 

. 7472 

. 2528 

37 

28 

36 

24 

. 6497 

. 3503 

.7700 

. 8471 

.4605 

.2119 

. 7489 

. 2511 

36 

24 

40 

25 

.56521 

.43479 

1.7693 

.68514 

1.4595 

1.2122 

.17505 

.82495 

35 

20 

44 

26 

. 6545 

. 3455 

.7685 

. 8557 

.4586 

.2124 

. 7521 

. 2478 

34 

16 

48 

27 

. 6769" 

. 3431 

.7678 

. 8600 

.4577 

.2127 

. 7538 

. 2462 

33 

12 

52 

28 

. 6593 

. 3407 

.7670 

. 8642 

.4568 

.2129 

. 7554 

. 2445 

32 

8 

56 

29 

. 6617 

. 3383 

.7663 

. 8685 

.4559 

.2132 

. 7571 

. 2429 

31 

4 

18 

30 

.56641 

.43359 

1.7655 

.68728 

1.4550 

1.2134 

.17587 

.82413 

30 

48 

4 

31 

. 6664 

. 3335 

.7648 

. 8771 

.4541 

.2136 

. 7604 

. 2396 

29 

56 

8 

32 

. 6688 

. 3311 

.7640 

. 8814 

.4532 

.2139 

. 7620 

. 2380 

28 

52 

12 

33 

. 6712 

. 3287 

.7633 

. 8857 

.4523 

.2141 

. 7637 

. 2363 

27 

48 

16 

31 

. 6736 

. 3263 

.7625 

. 8899 

.4514 

.2144 

. 7653 

. 2347 

26 

44 

20 

35 

.56760 

.43239 

1.7618 

.68942 

1.4505 

1.2146 

.17670 

.82330 

25 

40 

24 

36 

. 6784 

. 3216 

.7610 

. 8985 

.4496 

.2149 

. 7686 

. 2314 

21 

36 

28 

37 

. 6808 

. 3192 

.7603 

. 9028 

.4487 

.2151 

. 7703 

. 2297 

23 

32 

32 

38 

. 6832 

. 3168 

.7596 

. 9071 

.4478 

.2153 

. 7719 

. 2280 

22 

28 

36 

39 

. 6856 

. 3144 

.7588 

. 9114 

.4469 

.2156 

. 7736 

. 2264 

21 

24 

40 

40 

.56880 

.43120 

1.7581 

.69157 

1.4460 

1.2158 

.17752 

.82247 

20 

20 

44 

41 

. 6904 

. 3096 

.7573 

. 9200 

.4451 

.2161 

. 7769 

. 2231 

19 

16 

48 

42 

. 6928 

. 3072 

.7566 

. 9243 

.4442 

.2163 

. 7786 

. 2214 

18 

12 

52 

43 

. 6952 

. 3048 

.7559 

. 9286 

.4433 

.2166 

. 7802 

. 2198 

17 

8 

56 

44 

. 6976 

. 3024 

.7551 

. 9329 

.4424 

.2168 

. 7819 

. 2181 

16 

4 

19 

45 

.57000 

.43000 

1.7544 

.69372 

1.4415 

1.2171 

.17835 

.82165 

15 

41 

4 

46 

. 7023 

. 2976 

.7537 

. 9415 

.4406 

.2173 

. 7852 

. 2148 

14 

56 

8 

47 

. 7047 

. 2952 

.7529 

. 9459 

.4397 

.2175 

. 7868 

. 2131 

13 

52 

12 

48 

. 7071 

. 2929 

.7522 

. 9502 

.4388 

.2178 

. 7885 

. 2115 

12 

48 

16 

49 

. 7095 

. 2905 

.7514 

. 9545 

.4379 

.2180 

. 7902 

. 2098 

11 

44 

20 

50 

.57119 

.42881 

1.7507 

.69588 

1.4370 

1.2183 

.17918 

.82082 

10 

40 

24 

51 

. 7143 

. 2857 

.7500 

. 9631 

.4361 

.2185 

. 7935 

. 2065 

9 

36 

28 

52 

. 7167 

. 2833 

.7493 

. 9674 

.4$32 

.2188 

. 7951 

. 2048 

8 

32 

32 

53 

. 7191 

. 2S09 

.7485 

. 9718 

.4843 

.2190 

. 7968 

. 2032 

7 

28 

36 

54 

. 7214 

. 2785 

.7478 

. 9761 

.4335 

.2193 

. 7985 

. 2015 

6 

24 

40 

55 

.57238 

.42761 

1.7471 

.69804 

1.4326 

1.2195 

.18001 

.81998 

5 

20 

41 

56 

. 7262 

. 2738 

.7463 

. 9847 

.4317 

.2198 

. 8018 

. 1982 

4 

16 

48 

57 

. 7286 

. 2714 

.7456 

. 9891 

.4308 

.2200 

. 8035 

. 1965 

3 

12 

52 

58 

. 7310 

. 2690 

.7449 

. 9934 

.4299 

.2203 

. 8051 

. 1948 

2 

8 

56 

59 

. 7334 

. 2666 

.7442 

. 9977 

.4290 

.2205 

. 8068 

. 1932 

1 

4 

30 

60 

. 7358 

. 2642 

.7434 

.70021 

.4281 

.2208 

. 8085 

. 1915 

0 

40 

M. S. 

M 

Cosine. | 

Vis. Sin. 

Secante. 

Cotang. ■ 

Taugent. 

Cosec’ut 

Vrs.Cos 

Sine. 

M 

M.S. 

8 h 

124 c 




Natural. 



55° 

3 h 



































244 Natural Lines. 


2 h 

35 c 

Natural Trigonometrical Functions 

144° 


M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

20 

0 

.57358 

.42642 

1.7434 

.70021 

1.4281 

1.2208 

.18085 

.81915 

60 

TO 

4 

1 

. 7381 

. 2618 

.7427 

. 0064 

.4273 

.2210 

. 8101 

. 1898 

59 

56 

8 

2 

. 74U5 

. 2595 

.7120 

. 0107 

.4264 

.2213 

. 8118 

. 1882 

58 

52 

12 

3 

. 7429 

. 2571 

.7413 

. 0151 

.4255 

.2215 

. 8135 

. 1865 

57 

48 

16 

4 

. 7453 

. 2547 

.7405 

. 0194 

.4246 

.2218 

. 8151 

. 1848 

56 

44 

20 

5 

.57477 

.42523 

1.7398 

.70238 

1.4237 

1.2220 

.18168 

.81832 

55 

40 

24 

6 

. 7500 

. 2499 

.7391 

. 0281 

.4228 

.2223 

. 8185 

. 1815 

54 

30 

28 

7 

. 7524 

. 2476 

.7384 

. 0325 

.4220 

.2225 

. 8202 

. 1798 

53 

32 

32 

8 

. 7548 

. 2452 

.7377 

. 0368 

.4211 

.2228 

. 8218 

. 1781 

52 

28 

36 

9 

. 7572 

. 2428 

.7369 

. 0412 

.4202 

.2230 

. 8235 

. 1765 

51 

24 

40 

10 

.57596 

.42404 

1.7362 

.70455 

1.4193 

1.2233 

.18252 

.81748 

50 

20 

44 

11 

. .7619 

. 2380 

.7355 

. 0499 

.4185 

.2235 

. 8269 

. 1731 

49 

16 

48 

12 

. 7643 

. 2357 

.7348 

. 0542 

.4176 

.2238 

. 8285 

. 1714 

48 

12 

52 

13 

. 7667 

. 2333 

.7341 

. 0586 

.4167 

.2240 

. 8302 

. 1698 

47 

8 

56 

14 

. 7691 

. 2309 

.7334 

. 0629 

.4158 

.2243 

. 8319 

. 1681 

46 

4 

21 

15 

.57714 

.42285 

1.7327 

.70673 

1.4150 

1.2245 

.18336 

.81664 

45 

39 

4 

16 

. 7738 

. 2262 

.7319 

. 0717 

.4141 

.2248 

. 8353 

. 1647 

44 

50 

8 - 

17 

. 7762 

. 2238 

.7312 

. 0760 

.4132 

.2250 

. 8369 

. 1630 

43 

52 

12 

18 

. 7786 

. 2214 

.7305 

. 0804 

.4123 

.2253 

. 8386 

. 1614 

42 

48 

16 

19 

. 7809 

2190 

.7298 

. 0848 

.4115 

.2255 

. 8403 

. 1597 

41 

44 

20 

20 

.57833 

.42167 

1.7291 

.70891 

1.4106 

1.2258 

.18420 

.81580 

40 

40 

24 

21 

. 7857 

. 2143 

.7284 

. 0935 

.4097 

.2260 

. 8437 

. 1563 

39 

36 

28 

22 

. 7881 

. 2119 

.7277 

. 0979 

.4089 

.2263 

. 8453 

. 1546 

38 

32 

32 

23 

. 7904 

. 2096 

.7270 

. 1022 

.4080 

.2265 

. 8470 

. 1530 

37 

28 

36 

24 

. 7928 

. 2072 

.7263 

. 1066 

.4071 

.2268 

. 8487 

. 1513 

36 

24 

40 

25 

.57952 

.42048 

1.7256 

.71110 

1.4063 

1.2270 

.18504 

.81496 

35 

20 

44 

26 

. 7975 

. 2024 

.7249 

. 1154 

.4054 

.2273 

. 8521 

. 1479 

34 

16 

48 

27 

. 7999 

. 2001 

.7242 

. 1198 

.4045 

.2276 

. 8538 

. 1462 

33 

12 

52 

28 

. 8023 

. 1977 

.7234 

. 1241 

.4037 

.2278 

. 8555 

. 1445 

32 

8 

56 

29 

. 8047 

. 1953 

.7227 

. 1285 

.4028 

.2281 

. 8571 

. 1428 

31 

4 

22 

30 

.58070 

.41930 

1.7220 

.71329 

1.4019 

1.2283 

.18588 

.81411 

30 

38 

4 

31 

. 8094 

. 1906 

.7213 

. 1373 

.4011 

.2286 

. 8605 

. 1395 

29 

56 

8 

32 

. 8118 

. 1882 

.7206 

. 1417 

.4002 

.2288 

. 8622 

. 1378 

28 

52 

12 

33 

. 8141 

. 1859 

.7199 

. 1461 

.3994 

.2291 

. 8639 

. 1361 

27 

48 

16 

34 

. 8165 

. 1835 

.7192 

. 1505 

.39S5 

.2293 

. 8656 

. 1344 

26 

44 

20 

35 

.58189 

.41811 

1.7185 

.71549 

1.3976 

1.2296 

.18673 

.81327 

25 

40 

24 

36 

. 8212 

. 17S8 

.7178 

. 1593 

.3968 

.2298 

. 8690 

. 1310 

24 

36 

28 

37 

. 8236 

. 1764 

.7171 

. 1637 

.3959 

.2301 

. 8707 

. 1293 

23 

32 

32 

38 

. 8259 

. 1740 

.7161 

. 1681 

.3951 

.2304 

. 8724 

. 1276 

22 

28 

36 

39 

. 8283 

. 1717 

.7157 

. 1725 

.3942 

.2306 

. 8741 

. 1259 

21 

24 , 

40 

40 

.58307 

.41693 

1.7151 

.71769 

1.3933 

1.2309 

.18758 

.81212 

20 

20 

44 

41 

. 8330 

. 1669 

.7144 

. 1813 

.3925 

.2311 

. 8775 

. 1225 

19 

16 

48 

42 

. 8351 

. 1646 

.7137 

. 1857 

.3916 

.2314 

. 8792 

. 1208 

18 

12 

52 

43 

. 8378 

. 1622 

.7130 

. 1901 

.3908 

.2316 

. 8809 

. 1191 

17 

8 

56 

44 

. 8401 

. 1599 

.7123 

. 1945 

.3899 

.2319 

. 8826 

. 1174 

10 

4 

23 

45 

.58425 

.41575 

1.7116 

.71990 

1.3891 

1.2322 

.18843 

.81157 

15 

37 

4 

46 

. 8448 

. 1551 

.7109 

. 2034 

.3882 

.2324 

. 8860 

. 1140 

14 

56 

8 

47 

. 8472 

. 1528 

.7102 

. 2078 

.3874 

.2327 

. 8877 

. 1123 

13 

62 

12 

48 

. 8486 

. 1504 

.7095 

. 2122 

3865 

.2329 

. 8894 

. 1106 

12 

48 

16 

49 

. 8519 

. 1481 

.7088 

2166 

.3857 

.2332 

. 8911 

. 1089 

11 

44 

20 

50 

.58543 

.41457 

1.7081 

.72211 

1.3848 

1.2335 

.18.>28 

.81072 

10 

40 

24 

51 

. 8566 

. 1433 

.7075 

. 2255 

.3840 

.2337 

. 8945 

. 1055 

9 

30 

28 

52 

. 8590 

. 1410 

.7068 

. 2299 

.3831 

.2340 

. 8962 

. 1038 

8 

32 

32 

53 

. 8614 

. 1386 

.7061 

. 2344 

.3823 

.2342 

. 8979 

. 1021 

7 

28 

36 

54 

. 8637 

. 1363 

.7054 

. 2388 

.3814 

.2345 

. 8996 

. 1004 

6 

24 

40 

55 

.58661 

.41339 

1.7047 

.72432 

1.3806 

1.2348 

.19013 

.80987 

5 

20 

44 

56 

. 8684 

. 1316 

.7040 

. 2477 

.3797 

.2350 

. 9030 

. 0970 

4 

16 

48 

57 

. 8708 

. 1292 

.7033 

. 2521 

.3789 

.2353 

. 9047 

. 0953 

3 

12 

52 

58 

. 8731 

. 1268 

.7027 

. 2565 

.3781 

.2355 

. 9064 

. 0936 

2 

8 

56 

59 

. 8755 

. 1245 

.7020 

. 2610 

.3772 

.2358 

. 9081 

. 0919 

1 

4 

24 

60 

. 8778 

. 1221 

.7013 

. 2654 

.3701 

.2361 

. 9093 

. 0902 

0 

36 

M.S. 

M 

Cosine. 

Yrs. Sin. 

Secaute. 

Cotang.' 

Tangent. 

Cosec’nt 1 Vrs.Cos 

Sine. 

M 

M.S. 

8 h 

125 

D 



Natural. 




54°] 3 h 



























Natural Lines. 245 


2 h 

36 c 


Natural Trigonometrical Functions 

o 

CO 

rH 

9 h 

M.S. 

A1 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Seeante. 

iVrs. Sin 

1 Cosine. 

M 

M.S. 

34 

0 

•58778 

.41221 

1.7013 

.72654 

1.3764 

1.2361 

.19098 

.80902 

GO 

3G 

4 

1 

. 8802 

. 1198 

.7006 

. 2699 

.3755 

.2363 

. 9115 

. 0885 

59 

56 

8 

2 

. 8825 

. 1174 

.6999 

. -743 

.3747 

.2366 

. 9132 

. OS 67 

58 

52 

12 

3 

. 8849 

. 1151 

.6993 

. 2788 

.3738 

.2368 

. 9150 

. 0850 

57 

48 

16 

4 

. 8873 

. 1127 

.6986 

. 2832 

.3730 

.2371 

. 9167 

. 0833 

56 

44 

20 

5 

•58896 

.41104 

1.6979 

.72877 

1.3722 

1.2374 

.19184 

.80816 

55 

40 

24 

6 

• 8920 

. 10S0 

.6972 

. 2921 

.3713 

.2876 

. 9201 

. 0799 

54 

36 

28 

7 

• 8943 

. 1057 

.6965 

. 2966 

.3705 

.2379 

. 9218 

. 0782 

53 

32 

32 

8 

. 8967 

. 1033 

.6959 

. 3010 

.3697 

.2382 

. 9235 

. 0765 

52 

28 

36 

9 

. 8990 

. 1010 

.6952 

. 3055 

.3688 

.2384 

. 9252 

. 0747 

51 

24 

40 

10 

•59014 

.40986 

1.6945 

.73100 

1.3680 

1.2387 

.19270 

.80730 

50 

20 

44 

11 

. 9037 

. 0963 

.6938 

. 3144 

.3672 

.2389 

. 9287 

. 0713 

49 

16 

4S 

12 

. 9060 

. 0939 

.6932 

. 31S9 

.8663 

.2892 

. 9304 

. 0696 

48 

12 

52 

13 

. 90S4 

. 0916 

.6925 

. 3234 

.3655 

.2385 

. 9321 

. 0679 

47 

8 

56 

14 

. 9107 

. 0892 

.6918 

. 3278 

.3617 

.2397 

. 9338 

. 0662 

46 

4 

35 

15 

.59131 

.40869 

1.6912 

.73323 

1.3638 

1.2400 

. 19355 

.80644 

45 

35 

4 

16 

. 9154 

. 0845 

•6905 

. 3368 

.3630 

.240 1 

. 9373 

. 0027 

44 

56 

8 

17 

• 9178 

. 0822 

.6898 

. 3412 

.3622 

.2405 

. 9390 

. 0610 

43 

52 

12 

18 

. 9201 

. 0799 

.6891 

• 3457 

.3613 

.2408 

. 9407 

. 0593 

42 

48 

16 

19 

. 9225 

. 0775 

.6885 

• 1502 

.3605 

.2411 

. 9424 

. 0576 

41 

44 

20 

20 

.59248 

.40752 

1.6878 

■73547 

1.3597 

1.2413 

.19442 

.80558 

40 

40 

24 

24 

. 9272 

. 0728 

•6871 

. 3592 

.3588 

.2416 

. 9459 

. 0541 

89 

36 

28 

22 

. 9295 

. 0705 

.6865 

. 3637 

.3580 

.2419 

. 9476 

. 0524 

38 

32 

32 

23 

. 9318 

. 0681 

.6858 

. 3681 

.3572 

.2421 

. 9493 

. 0507 

37 

28 

36 

24 

. 9342 

. 0658 

.6851 

. 3726 

.3564 

.2424 

. 9511 

. 0489 

36 

24 

40 

25 

-59365 

.40635 

1.6845 

•73771 

1.3555 

1.2427 

.19528 

.80472 

35 

20 

44 

26 

. 9389 

. 0611 

.6838 

. 3816 

.3517 

.2429 

. 9545 

. 0455 

34 

16 

48 

27 

. 9412 

. 0588 

.6831 

. 3861 

.3539 

.2,432 

. 9562 

. 0437 

33 

12 

52 

2S 

. 9435 

. 0564 

.6825 

. 3906 

.8531 

.2435 

. 9580 

. 0420 

82 

8 

56 

29 

. 9459 

. 0541 

.6818 

. 3951 

.3522 

.2437 

. 9597 

. 0403 

31 

4 


30 

•594S2 

.40518 

1.6812 

.73996 

1.3614 

1.2440 

.19614 

.80386 

30 

34 

4 

31 

. 9506 

. 0494 

■6805 

. 4041 

.3506 

.2443 

. 9632 

. 0368 

•29 

56 

8 

32 

. 9529 

. 0471 

6798 

. 4086 

.3498 

.2445 

. 9649 

. 0351 

28 

52 

12 

33 

• 9552 

. 0447 

6792 

. 4131 

.3489 

.2448 

. 9666 

. 0334 

27 

48 

16 

34 

. 9576 

. 0424 

.6785 

. 4176 

.3481 

.2451 

. 9683 

. 0316 

26 

44 

20 

35 

.59599 

.40401 

1.6779 

•74221 

1.3473 

1.2 453 

.1970 L 

.80299 

25 

40 

24 

36 

. 9622 

. 0377 

6772 

. 4266 

.3465 

.2456 

. 9718 

. 0282 

24 

36 

28 

37 

. 9646 

. 0354 

.6766 

. 4312 

.3457 

.2459 

9736 

. 0264 

23 

32 

32 

38 

. 9669 

. 03.31 

.6759 

. 4357 

.3449 

.2161 

. 9753 

. 0247 

22 

28 

36 

39 

. 9692 

. 0307 

.6752 

. 4402 

.3440 

.2464 

. 9770 

. 0230 

21 

24 

40 

40 

.59716 

.40284 

1.6746 

.74447 

1.3432 

1.2467 

.19788 

.80212 

20 

20 

44 

41 

. 9739 

. 0261 

.6739 

. 4492 

.3424 

.2470 

. 9805 

. 0195 

19 

16 

48 

42 

. 9762 

. 0237 

.6733 

. 4538 

.3416 

.2472 

. 9822 

. 0177 

18 

12 

52 

43 

. 9786 

; 0214 

.6726 

. 4583 

.3108 

.2475 

. 9840 

. 0160 

17 

S 

56 

44 

. 9S09 

. 0191 

.6720 

. 4628 

.3400 

.2478 

. 9857 

. 0143 

16 

4 

37 

45 

.59832 

.40167 

1.6713 

.74673 

1.3392 

1.24MJ 

.19875 

.80125 

15 

33 

4 

46 

. 9856 

. 0144 

.6707 

. 4719 

.3383 

.2483 

. 9892 

. 0108 

14 

56 

8 

47 

. 9879 

. 0121 

.6700 

. 4764 

.3875 

.24 6 

. 9909 

. 0090 

13 

52 

12 

48 

. 9902 

. 0098 

.6694 

. 4809 

.33G7 

.248 s 

. 9927 

. 0073 

12 

48 

16 

49 

. 9926 

. 0074 

.6687 

. 4855 

.3359 

.2495 

. 9944 

. 0056 

11 

44 

20 

50 

.59949 

.40051 

1.6681 

.74900 

1.3351 

1.2494 

.19962 

.80088 

10 

40 

24 

51 

. 9972 

. 0028 

.6674 

. 4916 

.3343 

.2497 

. 9979 

. 0021 

9 

36 

28 

52 

. 9995 

. 0004 

.6668 

. 4991 

.3335 

.2499 

. 9997 

. 0003 

8 

32 

32 

53 

.60019 

.39981 

.6661 

. 5037 

.3327 

.2502 

.20014 

.79986 

7 

28 

36 

54 

. 0042 

. 9958 

.6655 

. 5082 

.3319 

.2505 

. 0031 

. 996S 

G 

24 

40 

55 

.60665 

.39935 

1.6648 

.75128 

1.3311 

1.2508 

.20049 

.79951 

5 

20 

44 

56 

. 0088 

. 9911 

.6642 

. 5173 

.3303 

.2510 

. 0066 

. 9933 

4 

16 

48 

57 

. 0112 

. 9888 

.6636 

. 5219 

.3294 

.2513 

. 0084 

. 9916 

3 

12 

52 

58 

. 0135 

. 9865 

.6629 

. 5214 

.3286 

.2516 

. 0101 

. 9898 

2 

8 

56 

59 

. 0158 

. 9842 

.6623 

. 5310 

.3278 

.2519 

. 0119 

. 9881 

i 

4 

38 

60 

. 0181 1 

. 9818 

.6616 

. 5355 

.3270 

.2521 

. 0136 

. 9863 

0 

32 

Af. S. 

M 

Cosine. I 

Vrs.Siu 

Secaute. 

Co tang. 

raiment. 

Cosec’nt. 

Vrs. Cos 

Sine. 

A1 

M.S. 

8 h 

126 c 




Natural. 



53° 

3 h 









































240 


Natural Links. 


'i h 

37' 


Natural Trigonometrical Functions 

142° 

9 h 

M.S. 

M 

Sine. 

Vrs. Cos 

Cosec’nte 

Tang. 

Cotang. 

Secante 

jVrs. Sin 

Cosine. 

M 

M.S. 


0 

.60181 

.39818 

1.6616 

.75365 

1.3270 

1.2521 

.20136 

.79863 

60 

32 

4 

1 

0205 

. 9795 

.6610 

. 5401 

.3262 

.2524 

|. 0154 

. 9846 

59 

56 

8 

2 

. 0228 

. 9772 

.6603 

. 5447 

.3254 

.2527 

. 0171 

. 9828 

68 

52 

12 

3 

. 0251 

. 9749 

.6597 

. 5492 

.3246 

.2530 

. 0189 

. 9811 

57 

48 

16 

4 

. 0274 

. 9726 

.6591 

. 5538 

.3238 

.2532 

. 0206 

. 9793 

56 

44 

20 

6 

.60298 

.39702 

1.6584 

.75584 

1.3230 

1.2535 

.20224 

.79776 

55 

40 

24 

6 

. 0320 

. 9679 

.6578 

. 5629 

.3222 

.2538 

. 0242 

. 9758 

54 

36 

28 

7 

. 0344 

. 9656 

.6572 

. 5675 

.3214 

.2541 

. 0259 

. 9741 

53 

32 

32 

8 

. 0367 

. 9633 

.6565 

. 5721 

.3206 

.2543 

. 0277 

. 9723 

52 

28 

36 

9 

. 0390 

. 9610 

.6559 

. 5767 

.3198 

.2546 

. 0294 

. 9706 

51 

24 

40 

10 

.60413 

.39586 

1.6552 

.75812 

1.3190 

1.25+9 

.20312 

.79688 

50 

20 

44 

11 

. 0437 

. 9563 

.6546 

. 5858 

.3182 

.2552 

. 0329 

. 9670 

49 

16 

48 

12 

. 0460 

. 9540 

.6540 

. 5904 

.3174 

.2554 

. 0347 

. 9653 

48 

12 

52 

13 

. 0483 

. 9517 

.6533 

. 5950 

.3166 

.2557 

. 0365 

. 9635 

47 

8 

56 

14 

. 0506 

. 9494 

.6527 

. 5996 

.3159 

.2560 

. 0382 

. 9618 

46 

4 

29 

15 

.60529 

.39471 

1.6521 

.76042 

1.3151 

1.2563 

.20400 

.79600 

45 

31 

4 

16 

. 0552 

. 9447 

.6514 

. 6088 

.3143 

.2565 

. 0417 

. 9582 

44 

56 

8 

17 

. 0576 

. 9424 

.6508 

. 6134 

.3135 

.2568 

. 0435 

. 9565 

43 

52 

12 

IS 

. 0599 

. 9404 

.6502 

. 6179 

.3127 

.2571 

. 6453 

. 9547 

42 

48 

16 

19 

. 0622 

. 937S 

.6496 

. 6225 

.3119 

.2574 

. 0470 

. 9530 

41 

44 

20 

20 

.69645 

.39355 

1.6489 

•76271 

1.3111 

1.2577 

.20488 

.79512 

40 

40 

24 

21 

. 0668 

. 9332 

.6483 

. 6317 

.3103 

.2579 

. 0505 

. 9494 

39 

36 

28 

22 

. 0691 

. 9309 

.6477 

. 6364 

.3095 

.2582 

. 0523 

. 9477 

38 

32 

32 

23 

. 0714 

. 9285 

.6470 

. 6410 

.3087 

.2585 

. 0541 

. 9459 

37 

28 

36 

24 

. 0737 

. 9262 

.6464 

. 6456 

3079 

.2588 

. 0558 

. 9441 

36 

24 

40 

25 

.60761 

.39239 

1.6458 

.76502 

1.3071 

1.2591 

.20576 

.79424 

35 

20 

44 

26 

• 0784 

. 9216 

.6452 

. 6548 

.3064 

.2593 

. 0594 

. 9406 

34 

16 

48 

27 

. 0807 

. 9193 

.6445 

. 6594 

.3056 

.2596 

. 0611 

. 9388 

S3 

12 

52 

2S 

. 0830 

. 9170 

.6439 

. 6640 

.3048 

.2599 

. 0629 

. 9371 

32 

8 

56 

29 

■ 0853 

. 9147 

.6433 

. 6686 

.3040 

.2602 

. 0647 

. 9353 

31 

4 

30 

30 

.60876 

.3912 4 

1.6427 

.76733 

1.3032 

1.2605 

.20665 

.79335 

30 

30 

4 

31 

. 0899 

. 9101 

.6420 

. 6779 

.3024 

.2607 

. 0682 

. 9318 

29 

56 

. 8 

32 

• 0922 

. 9078 

.6414 

. 6825 

.3016 

.2610 

. 0700 

. 9300 

28 

52 

12 

33 

. 0945 

. £055 

.6408 

. 6871 

.3009 

.2613 

. 0718 

. 9282 

27 

48 

16 

34 

• 0963 

. 9031 

.6402 

. 6918 

.3001 

.2616 

. 0735 

. 9264 

26 

44 

20 

35 

•60991 

.39008 

1.6396 

.76964 

1.2993 

1.2619 

.20753 

.79247 

25 

40 

24 

36 

. 1014 

. 8985 

.6389 

. 7010 

.2985 

.2622 

. 0771 

. 9229 

24 

36 

28 

37 

. 1037 

. 8962 

.6383 

. 7057 

.2977 

.2624 

0789 

. 9211 

23 

32 

32 

38 

• 1061 

. 8939 

.6377 

. 7103 

.2970 

.2627 

. 0806 

. 9193 

22 

28 

3G 

39 

• 1084 

. 8916 

.6371 

. 7149 

.2962 

.2630 

. 0824 

. 9176 

21 

24 

40 

40 

•61107 

.38893 

1.6365 

.771S6 

1.2954 

1.2633 

.20842 

.7915S 

20 

20 

44 

41 

. 1130 

. 8870 

.6359 

. 7242 

.2946 

.2636 

. 0860 

. 9140 

19 

16 

48 

42 

. 1153 

. 8847 

.6352 

. 7289 

.2938 

.2639 

. 0878 

. 9122 

18 

12 

52 

43 

. 1176 

. 8824 

.6346 

. 7335 

.2931 

.2641 

. 0835 

. 9104 

17 

8 

56 

44 

. 1199 

. 8801 

.6340 

. 7382 

.2923 

.2644 

. 0913 

. 9087 

16 

4 

31 

45 

.61222 

.38778 

1.6334 

.77428 

1.2915 

1.2647 

.20931 

.79069 

15 

29 

4 

46 

. 1245 

. 8755 

.6328 

. 7475 

.2907 

.2650 

. 0949 

. 9051 

14 

56 

8 

47 

. 1268 

. 8732 

.6322 

. 7521 

.2900 

.2653 

. 0967 

. 9033 

13 

52 

12 

48 

. 1290 

•. 8709 

.6316 

. 7568 

.2892 

.2656 

. 0984 

. 9015 

12 

48 

16 

49 

. 1314 

. 86S6 

.6309 

. 7614 

.2884 

.2659 

. 1002 

. 8998 

11 

44 

20 

50 

.61337 

.38663 

1.6303 

.77661 

1.2876 

1.2661 

.21020 

.78980 

10 

40 

24 

51 

. 1360 

. 8640 

.6297 

. 7708 

.2869 

.2664 

. 1038 

. 8962 

9 

36 

28 

52 

. 1383 

. 8617 

.6291 

. 7754 

.2861 

.2667 

. 1056 

. 8914 

8 

32 

32 

53 

. 1405 

. 8594 

.6285 

. 7801 

.2853 

.2670 

. 1074 

. 8926 

7 

28 

36 

54 

. 1428 

. 8571 

.6279 

. 7848 

.2845 

.2673 

. 1091 

. 8008 

6 

24 

40 

55 

.61451 

.38548 

1 6273 

.77895 

1.2838 

1.2676 

.21109 

.78860 

5 

20 

44 

56 

. 1474 

. 8525 

.6267 

. 7941 

.2830 

.2679 

. 1127 

. 8873 

4 

16 

48 

67 

. 1497 

. 8503 

.6261 

. 7988 

.2822 

.2681 

. 1145 

. 8855 

3 

12 

52 

58 

. 1520 

. 8480 

.6255 

. 8035 

.2815 

.2684 

. 1163 

. 8837 

2 

8 

56 

59 

. 1543 

. 8467 

.6249 

. 8082 

.2807 

.2687 

. 1181 

. 8S19 

i 

4 

32 

60 

. 1566 

. 8434 

.6243 

. 8128 

.2799 

.2690 

. 1199 

. 8801 

0 

28 

M.S. 

M 

Coslue. 

Vrs. Sin. 

Secante. 

Dofhug. 

Tangent. 

Cosecant.] 

Vrs. Cos 

Sine. 

M 

M.S. 

8 h 

127° 



Natural. 



52°j 

3 h 































Natural Lines. 


24? 


2 h 

38 c 

Natural Trigonometrical Functions 

141° 

9 h 

M.S. 

M 

Sine. 

Yrs.Cos 

Cosee'nte 

Tang. 

Cotang. 

Secante 

Vrs.Sin 

Cosine. 

M 

M.S. 

3^4 

0 

.61566 

.38434 

1.6243 

.78128 

1.2799 

1.2690 

.21199 

.78801 

60 

as 

4 

1 

. 1589 

. 8411 

.6237 

. 8175 

.2792 

.2693 

. 1217 

. 8783 

59 

56 

8 

2 

. 1612 

. 83S8 

.6231 

. 8222 

.2784 

.2696 

. 1235 

. 8765 

58 

52 

12 

3 

. 1635 

. 8365 

.6224 

. 8269 

.2776 

.2699 

. 1253 

. 8747 

57 

48 

16 

4 

. 1658 

. 8342 

.6218 

. 8316 

.2769 

.2702 

. 1271 

. 8729 

56 

44 

20 

5 

.61681 

.38319 

1.6212 

.78363 

1.2761 

1.2705 

.21288 

.78711 

55 

40 

24 

6 

. 1703 

. 8296 

.6206 

. 8410 

.2753 

.2707 

. 1306 

. 8693 

54 

36 

28 

7 

. 1726 

. 8273 

.6200 

. 8457 

.2746 

.2710 

. 1324 

. 8675 

53 

32 

32 

8 

. 1749 

. 8251 

.6194 

. 8504 

.2738 

.2713 

. 1342 

. 8657 

52 

28 

36 

9 

. 1772 

. 8228 

.0188 

. 8551 

.2730 

.2716 

. 1360 

. 8640 

51 

24 

40 

10 

.61795 

.38205 

1.6182 

.78598 

1.2723 

1.2719 

.21378 

.78622 

■50 

20 

44 

11 

. 1818 

. 8182. 

.6176 

. 8645 

.2715 

.2722 

. 1396 

. 8604 

49 

16 

48 

12 

. 1841 

. 8159 

.6170 

. 8692 

.2708 

.2725 

. 1414 

. 8586 

48 

12 

52 

13 

. 1864 

. 8136 

.6164 

. 8739 

.2700 

.2728 

. 1432 

. 8568 

47 

8 

56 

14 

. 1886 

. 8113 

.6159 

. 8786 

.2692 

.2731 

. 1450 

. 8550 

46 

4 

33 

15 

.61909 

.38091 

1.6153 

.78834 

1.2685 

1.2734 

.21468 

.78532 

45 

27 

4 

16 

. 1932 

. 8068 

.6147 

. 8881 

.2677 

.2737 

. 1486 

. 8514 

44 

56 

8 

17 

. 1955 

. 8045 

.6141 

. 8928 

.2670 

.2739 

. 1504 

. 8496 

43 

52 

12 

18 

. 1978 

. 8022 

.6135 

. 8975 

.2662 

.2742 

. 1522 

. 8478 

42 

48 

16 

19 

. 2001 

. 7999 

.6129 

. 9022 

.2655 

.2745 

. 1540 

. 8460 

41 

44 

20 

20 

.62023 

.37976 

1.6123 

.79070 

1.2647 

1.2748 

.2155S 

.78441 

40 

40 

24 

21 

. 2046 

. 7954 

.6117 

. 9117 

.2639 

.2751 

. 1576 

. 8423 

39 

36 

28 

22 

. 2069 

. 7931 

.6111 

. 9164 

.2632 

.2754 

. 1594 

. 8405 

38 

32 

32 

23 

. 2092 

. 7908 

.6105 

. 9212 

.2624 

.2757 

. 1612 

. 8387 

37 

28 

36 

24 

. 2115 

. 7885 

.6099 

. 9259 

.2617 

.2760 

. 1631 

. 8369 

36 

24 

40 

25 

.62137 

.37862 

1.6093 

.79306 

1.2609 

1.2763 

.21649 

.78351 

35 

20 

44 

26 

. 2160 

. 7840 

.6087 

. 9354 

.2602 

.2766 

. 1667 

. 8333 

34 

16 

48 

27 

. 2183 

. 7817 

.6081 

. 9401 

.2594 

.2769 

. 1685 

. 8315 

33 

12 

52 

28 

. 2206 

. 7794 

.6077 

. 9449 

.2587 

.2772 

. 1703 

. 8297 

32 

8 

56 

29 

. 2229 

. 7771 

.6070 

. 9496 

.2579 

.2775 

. 1721 

. 8279 

31 

4 

34 

30 

.62251 

.37748 

1.6064 

.79543 

1.2572 

1.2778 

.21739 

.78261 

30 

26 

4 

31 

. 2274 

. 7726 

.6058 

. 9591 

.2564 

.2781 

. 1757 

. 8243 

29 

•66 

8 

32 

. 2297 

. 7703 

.6052 

. 9639 

.2557 

.2784 

. 1775 

. 8224 

28 

52 

12 

33 

. 2320 

. 7680 

.6046 

. 9686 

.2549 

.2787 

. 1793 

. 8206 

27 

48 

16 

34 

. 2342 

. 7657 

.6040 

. 9734 

.2542 

.2790 

. 1812 

. 8188 

26 

44 

20 

35 

.62365 

.37635 

1.6034 

.79781 

1.2534 

1.2793 

.21830 

.78170 

25 

40 

24 

36 

. 2388 

. 7612 

.6029 

. 9829 

.2527 

.2795 

. 1848 

. 8152 

24 

36 

28 

37 

. 2411 

. 7589 

.6023 

. 9876 

.2519 

.2798 

. 1866 

. 8134 

23 

32 

32 

38 

. 2433 

. 7566 

.6017 

. 9924 

.2512 

.2801 

. 1884 

. 8116 

22 

28 

36 

39 

. 2456 

. 7544 

.6011 

. 9972 

.2504 

.2804 

. 1902 

. 8097 

21 

24 

40 

40 

.62479 

.37521 

1.6005 

.80020 

1.2497 

1.2807 

.21921 

.78079 

20 

20 

44 

41 

. 2501 

. 7498 

.6000 

. 0067 

.2489 

.2810 

. 1939 

. 8061 

19 

16 

48 

42 

. 2524 

. 7476 

.5994 

. 0115 

.2482 

.2813 

. 1957 

. 8043 

18 

12 

52 

43 

. 2547 

. 7453 

.5988 

. 0163 

.2475 

.2816 

. 1975 

. 8025 

17 

8 

56 

44 

. 2570 

. 7430 

.5982 

. 0211 

.2467 

.2819 

. 1993 

. 8007 

16 

4 

35 

45 

.62592 

.37408 

1.5976 

.80258 

1.2460 

1.2822 

.22011 

.7798S 

15 

25 

4 

46 

. 2615 

. 7385 

.5971 

. 0306 

.2452 

.2825 

. 2030 

. 7970 

14 

56 

8 

47 

. 2638 

. 7362 

.5965 

. 0354 

.2445 

.2828 

. 2048 

. 7952 

13 

52 

12 

48 

. 2660 

. 7340 

.5959 

. 0402 

.2437 

.2831 

. 2066 

. 7934 

12 

48 

16 

49 

. 2683 

. 7317 

.5953 

. 0450 

.2430 

.2834 

. 2084 

. 7915 

11 

44 

20 

50 

.62706 

.37294 

1.5947 

.80498 

1.2423 

1.2837 

.22103 

.77897 

10 

40 

24 

51 

. 2728 

. 7272 

.5942 

. 0546 

.2415 

.2840 

. 2121 

. 7879 

9 

36 

28 

52 

. 2751 

. 7249 

.5936 

. 0594 

.2408 

.2843 

. 2139 

. 7861 

8 

32 

32 

53 

. 2774 

. 7226 

.5930 

. 0642 

.2400 

.2846 

. 2157 

. 7842 

7 

28 

36 

54 

. 2796 

. 7204 

.5924 

. 0690 

.2393 

.2849 

. 2176 

. 7824 

6 

24 

40 

55 

.62819 

.37181 

1.5919 

.80738 

1.2386 

1.2852 

.22194 

.77806 

5 

20 

44 

56 

. 2841- 

. 7158 

.5913 

. 0786 

.2378 

.2855 

. 2212 

. 7788 

4 

16 

4S 

57 

. 2864 

. 7136 

.5907 

. 0834 

.2371 

.2858 

. 2230 

. 7769 

3 

12 

52 

58 

. 2887 

. 7113 

.5901 

. 0882 

.2364 

.2861 

. 2249 

. 7751 

2 

8 

56 

59 

. 2909 

. 7090 

.5896 

. 0930 

.2356 

.2864 

. 2267 

. 7733 

1 

4 

30 

60 

. 2932 

. 7068 I 

.5890 

. 0978 

.2349 

.2867 

. 2285 

. 7715 

0 

24: 

M. S. 

M 

Cosine. 

Vrs.Sin.l 

Secante. 

CotaugjTaugent. 

Coseu’ntl 

Vrs.Cos 1 

Sine. | 

M 

vr.s. 

A 

00 

128 c 




Natural. 


fc - -- 

51° 

3 h 












































248 


Natural Lines. 


Oh 

At 

39° 

Natural Trigonometrical Functions 

140° 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

| Tang. 

Cotang. 

Secaute. 

Vrs. Sin 

1 Cosine. 

M 

M.S. 

30 

0 

.62932 

.37068 

1.5890 

.80978 

1.2349 

1.2867 

.22285 

.77715 

60 

2<t 

4 

1 

. 2955 

. 7045 

.5884 

. 1026 

.2342 

.2871 

. 2304 

. 7696 

59 

56 

8 

2 

. 2977 

. 7023 

.5879 

. 1075 

.2334 

.2874 

. 2322 

. 7678 

58 

52 

12 

3 

. 3000 

. 7000 

.5873 

. 1123 

.2327 

.2877 

. 2340 

. 7660 

57 

48 

16 

4 

. 3022 

. 6977 

.5867 

. 1171 

.2320 

.2880 

. 2359 

. 7641 

56 

44 

20 

5 

.63045 

.36955 

1.5862 

.81219 

1.2312 

1.2883 

.22377 

.77623 

55 

40 

24 

6 

. 3067 

. 6932 

.5856 

. 1268 

.2305 

.2886 

. 2395 

. 7605 

54 

36 

28 

7 

. 3090 

. 6910 

.5850 

. 1316 

.2297 

.2389 

. 2414 

. 7586 

53 

32 

32 

8 

. 3113 

. 6887 

.5845 

. 1364 

.2290 

.2892 

. 2432 

. 7568 

52 

28 

36 

9 

. 3135 

. 6865 

.5839 

. 1413 

.2283 

.2895 

. 2450 

. 7549 

51 

24 

40 

10 

.63158 

.36842 

1.5833 

.81461 

1.2276 

1.2898 

.22469 

.77531 

50 

20 

44 

11 

. 3180 

. 6820 

.5828 

. 1509 

.2268 

.2901 

. 24S7 

. 7513 

49 

16 

48 

12 

. 3203 

. 6797 

.5822 

. 1558 

.2261 

.2904 

. 2505 

. 7494 

48 

12 

52 

13 

. 3225 

. 6774 

.5816 

. 1606 

.2254 

.2907 

. 2524 

. 7476 

47 

8 

50 

14 

. 3248 

. 6752 

.5811 

. 1655 

.2247 

.2910 

. 2542 

. 7458 

46 

4 

37 

15 

.63270 

.36729 

1.5805 

.81703 

1.2239 

1.2913 

.22561 

.77439 

45 

23 

4 

16 

. 3293 

. 6707 

.5799 

. 1752 

.2232 

.2916 

. 2579 

. 7421 

44 

56 

8 

17 

. 3315 

. 6684 

.5794 

. 1800 

.2225 

.2919 

. 2597 

. 7402 

43 

52 

12 

18 

. 3338 

. 6662 

.5788 

. 1849 

.2218 

.2922 

. 2616 

. 7334 

42 

48 

16 

19 

. 3360 

6639 

.5783 

. 1898 

.2210 

.2926 

. 2634 

. 7365 

41 

44 

20 

20 

.63383 

.36617 

1.5777 

.81946 

1.2203 

1.2929 

.22653 

.77347 

40 

40 

24 

21 

. 3405 

. 6594 

.5771 

. 1995 

.2196 

.2932 

. 2671 

. 7329 

39 

36 

28 

22 

. 3428 

. 6572 

.5766 

. 2043 

.2189 

.2935 

. 2690 

. 7310 

38 

32 

32 

23 

. 3450 

. 6549 

•57 60 

. 2092 

.2181 

.2938 

. 2708 

. 7292 

37 

28 

36 

24 

. 3473 

. 6527 

.5755 

. 2141 

.2174 

.2941 

. 2727 

. 7273 

36 

24 

40 

25 

.63495 

.36504 

1.5749 

.82190 

1.2167 

1.2944 

.22745 

.77255 

35 

20 

44 

26 

. 3518 

. 6482 

.5743 

. 2238 

.2160 

.2947 

. 2763 

. 7236 

34 

16 

48 

27 

. 3540 

. 6459 

■5738 

. 2287 

.2152 

.2950 

. 2782 

. 7218 

33 

12 

52 

28 

. 3563 

. 6437 

.5732 

. 2336 

.2145 

.2953 

. 2800 

. 7199 

32 

8 

56 

29 

. 3585 

. 6415 

.5727 

. 2385 

.2133 

.2956 

. 2819 

. 7181 

31 

4 

38 

30 

.63608 

.36392 

1.5721 

.82434 

1.2131 

1.2960 

.22837 

.77162 

30 

22 

4' 

31 

. 3630 

. 6370 

.5716 

. 2482 

.2124 

.2963 

. 2856 

. 7144 

29 

56 

8 

32 

. 3653 

. 6347 

.5710 

. 2531 

.2117 

.2966 

. 2874 

. 7125 

28 

52 

12 

33 

. 3675 

. 6325 

.5705 

. 2580 

.2109 

.2969 

. 2893 

. 7107 

27 

48 

16 

34 

. 3697 

. 6302 

.5699 

. 2629 

.2102 

.2972 

. 2912 

. 7088 

26 

44 

20 

35 

.63720 

.36280 

1.5694 

.82678 

1.2095 

1.2975 

.22930 

.77070 

25 

40 

24 

36 

. 3742 

. 6258 

.5088 

. 2727 

.2088 

.2978 

. 2949 

. 7051 

24 

36 

28 

37 

. 3765 

. 6235 

.5683 

. 2776 

.2081 

.2981 

. 2967 

. 7033 

23 

32 

32 

38 

. 3787 

. 6213 

.5677 

. 2825 

.2074 

.2985 

. 2986 

. 7014 

22 

28 

36 

39 

. 3810 

. 6190 

.5672 

. 2874 

.2066 

.2988 

. 3004 

. 6996 

21 

24 

40 

40 

.63832 

.36168 

1.5666 

.82923 

1.2059 

1.2a91 

.23023 

.76977 

20 

20 

44 

41 

. 3S54 

. 6146 

.5.661 

. 2972 

.2052 

.2994 

. 3041 

. 6958 

19 

16 

48 

42 

. 3877 

. 6123 

.5655 

. 3022 

.2045 

.2997 

. 3060 

. 6940 

18 

12 

52 

43 

. 3899 

. 6101 

.5650 

. 3071 

.2038 

.3000 

. 3079 

. 6921 

17 

8 

56 

44 

. 3021 

. 6078 

.5644 

. 3120 

.2031 

.3003 

. 3097 

. 6903 

16 

4 

30 

45 

.63944 

.36056 

1.5639 

.83169 

1.2024 

1.3006 

.23116 

.76884 

15 

21 

1 

46 

. 3966 

. 6034 

.5633 

. 3218 

.2016 

.3010 

. 3134 

. 6865 

14 

56 

8 

47 

. 3989 

. 6011 

.5628 

. 3267 

2009 

.3013 

. 3153 

. 6847 

13 

52 

12 

48 

. 4011 

. 59S9 

.5622 

. 3317 

.2002 

.3016 

. 3172 

. 6S28 

12 

48 

16 

49 

. 4033 

. 5967 

.5617 

. 3366 

.1995 

.3019 

. 3190 

6810 

11 

44 

20 

50 

.64056 

.35944 

1.5611 

.83415 

1.1988 

1.3022 

.232U9 

.76791 

10 

40 

24 

51 

. 4078 

. 5922 

.5606 

. 3465 

.1981 

.3025 

. 3227 

. 6772 

9 

36 

2S 

52 

. 4100 

. 5900 

.5600 

. 3514 

.1974 

.3029 

. 3246 

. 6754 

8 

32 

32 

53 

. 4123 

. 5877 

.5595 

. 3563 

.1967 

.3032 

. 3265 

. 6735 

7 

28 

36 

54 

. 4145 

. 5855 

.5590 

. 3613 

.1960 

.3035 

. 3283 

. 6716 

6 

24 

40 

55 

.64167 

.35833 

1.5584 

.83662 

1.1953 

1.3038 

.23302 

.76698 

5 

20 

44 

56 

. 4189 

. 5810 

.5579 

. 3712 

.1946 

.3041 

. 3321 

. 6679 

4 

16 

48 

57 

. 4212 

. 578S 

.5573 

. 3761 

.1939 

.3044 

. 3339 

. 6660 

3 

12 

52 

58 

. 4234 

.• 5766 

.5568 

. 3811 

.1932 

.3048 

. 3358 

. 6642 

2 

8 

56 

59 

. 4256 

. 5743 

.5563 

. 3860 

.1924 

.3051 

. 3377 

. 6623 

1 

4 

40 

60 

. 4279 

. 5721 

.5557 

. 3910 

.1917 

.3054 

. 3395 

6604 

0 

20 

M.S. 

M 

Cosine. 

Yrs.Siu.l Secante. 

Cotang. 

Tangent. 

Cosec’nt 

V rs. Cos 

Sine. 

M 

M.S. 

8 h 

129 ( 




Natural. 




50° 

3 h 

























Natural Lines. 


249 


2- 

o 

o 

Natural Trigonometrical Functions 

139° 

9 h 

M.S. 

M 

Sine. 

jVrs.Cos 

"Cosec’nte 

I Tang. 

Cotang 

Secante 

Yrs.Sin 

Cosine. 

M 

M.S. 

40 

0 

.64279 

.35721 

1.5557 

| .83910 

1.1917 

1.3054 

.23395 

.70604 

60 

20 

4 

X 

. 4301 

. 5699 

.5552 

. 3959 

.1910 

.3057 

. 3414 

. 6586 

59 

56 

8 

2 

. 4323 

. 5677 

.5546 

. 4009 

.1903 

.3060 

. 3433 

. 6567 

58 

52 

12 

3 

. 4345 

. 5654 

.5541 

. 4059 

.1896 

.3064 

. 3452 

. 6548 

57 

48 

10 

4 

. 430S 

. 6632 

.5536 

. 4108 

.1889 

.3067 

. 3470 

. 6530 

56 

44 

20 

5 

.64390 

.35610 

1.5530 

.84158 

1.1882 

1.3070 

.23489 

.76511 

55 

40 

24 

6 

. 4412 

. 5588 

.5525 

. 4208 

.1875 

.3073 

. 3508 

. 6492 

54 

36 

28 

7 

. 4435 

. 5565 

.5520 

. 4257 

.1868 

.3076 

. 3527 

. 6473 

53 

32 

82 

8 

. 4457 

. 5543 

.5514 

. 4307 

.1861 

.3080 

. 3545 

. 6455 

52 

28 

36 

9 

. 4479 

. 5521 

.5509 

. 4357 

.1854 

.3083 

. 3564 

. 6436 

51 

24 

40 

10 

.64501 

.35499 

1.5503 

.84407 

1.1847 

1.3086 

.23583 

.76417 

50 

20 

41 

11 

. 4523 

. 5476 

.5498 

. 4457 

.1840 

.3089 

. 3602 

. 6398 

49 

16 

48 

12 

. 4546 

. 5454 

.5493 

. 4506 

.1833 

.3092 

. 3620 

. 6380 

48 

12 

52 

13 

. 4568 

. 5432 

.5487 

. 4556 

.1826 

.3096 

. 3039 

. 6361 

47 

8 

56 

14 

. 4590 

. 5410 

.5482 

. 4606 

.1819 

.3099 

. 3658 

. 6342 

46 

4 

41 

15 

.64612 

.35388 

1.5477 

.84656 

1.1812 

1.3102 

.23677 

.76323 

45 

19 

4 

16 

. 4635 

. 5365 

.5471 

. 4706 

.1805 

.3105 

. 3695 

. 6304 

44 

56 

8 

17 

. 4657 

. 5343 

.5466 

. 4756 

.1798 

.3109 

. 3714 

. 6286 

43 

52 

12 

18 

. 4079 

. 5321 

.5461 

. 4806 

.1791 

.3112 

. 3733 

. 6267 

42 

48 

16 

19 

. 4701 

. 5299 

.5456 

. 4856 

.1785 

.3115 

. 3752 

. 6248 

41 

44 

20 

20 

.64723 

.35277 

1.5450 

.84906 

1.1778 

1.3118 

.23771 

.76229 

40 

40 

24 

21 

. 4745 

. 5254 

.5445 

. 4956 

.1771 

.3121 

. 3790 

. 6210 

39 

36 

28 

22 

. 4768 

. 5232 

.5440 

. 5006 

.1764 

.3125 

. 3808 

. 6191 

38 

32 

32 

23 

. 4790 

. 5210 

.5434 

. 5056 

.1757 

.3128 

. 3827 

. 6173 

37 

28 

36 

24 

. 4812 

. 5188 

.5429 

. 5107 

.1750 

.3131 

. 3846 

. 6154 

36 

24 

40 

25 

.64834 

.35166 

1.5424 

.85157 

1.1743 

1.3134 

.23865 

.76135 

35 

20 

44 

26 

. 4856 

. 5144 

.5419 

. 5207 

.1736 

.3138 

. 3884 

. 6116 

34 

16 

48 

27 

. 4878 

. 5121 

.5413 

. -5257 

.1729 

.3141 

. 3903 

. 6097 

33 

12 

52 

28 

. 4900 

. 5099 

.5408 

. 5307 

.1722 

.3144 

. 3922 

. 6078 

32 

8 

56 

29 

. 4923 

. 5077 

.5403 

. 5358 

.1715 

.3148 

. 3940 

. 6059 

31 

4 

42 

30 

.64945 

.35055 

1.5398 

.85408 

1.1708 

1.3151 

.23959 

.76041 

30 

18 

4 

31 

. 4967 

. 5033 

.5392 

. 5458 

.1702 

.3154 

. 3978 

. 6022 

29 

56 

8 

32 

. 4989 

. 5011 

.5387 

. 5509 

.1695 

.3157 

. 3997 

. 6003 

28 

52 

12 

33 

. 5011 

. 4989 

.5382 

. 5559 

.1688 

.3161 

. 4016 

. 5984 

27 

48 

16 

34 

. 5033 

. 4967 

.5377 

. 5609 

.1681 

.3164 

. 4035 

. 5965 

26 

44 

20 

35 

.65055 

.34945 

1.5371 

.85660 

1.1674 

1.3167 

.24054 

.75946 

25 

40 

24 

36 

. 5077 

. 4922 

.5366 

. 5710 

.1667 

.3170 

. 4073 

. 5927 

24 

36 

28 

37 

. 5099 

. 4900 

.5361 

. 5761 

.1660 

.3174 

. 4092 

. 5908 

23 

32 

32 

38 

. 5121 

. 4878 

.5356 

. 5811 

.1653 

.3177 

. 4111 

. 5839 

22 

28 

36 

39 

. 5144 

. 4856 

.5351 

. 5862 

.1647 

.3180 

. 4130 

. 6870 

21 

24 

40 

40 

.65166 

.34834 

1.5345 

.85912 

1.1640 

1.3184 

.24149 

.75851 

20 

20 

44 

41 

. 5188 

. 4812 

.5340 

. 5963 

.1633 

.3187 

. 4168 

. 5832 

19 

16 

48 

42 

. 5210 

. 4-790 

.5335 

. 6013 

.1626 

.3190 

. 4186 

. 5813 

18 

12 

52 

43 

. 5232 

. 4768 

.5330 

. 6064 

.1619 

.3193 

. 4205 

. 5794 

17 

8 

56 

44 

. 5254 

. 4740 

.5325 

. 6115 

.1612 

.3197 

. 4224 

. 6775 

16 

4 

43 

45 

.6527 6 

.34724 

1.5319 

.86165 

1.1605 

1.3200 

.24243 

.75756 

15 

17 

4 

46 

. 5298 

. 4702 

.5314 

. 6216 

.1599 

.3203 

. 4262 

. 5737 

14 

56 

8 

47 

. 5320 

. 4680 

.5309 

. 6267 

.1592 

.3207 

. 4281 

. 5718 

13 

52 

12 

48 

. 5342 

. 4658 

.5304 

. 6318 

.1585 

.3210 

. 4300 

. 5699 

12 

48 

16 

49 

. 5364 

. 4636 

.5299 

. 6368 

.1578 

.3213 

. 4319 

. 5680 

11 

44 

20 

50 

.65386 

.34614 

1.5294 

.86419 

1.1571 

1.3217 

.24338 

.75661 

10 

40 

24 

51 

. 5408 

. 4592 

.5289 

. 6470 

.1565 

.3220 

. 4357 

. 5042 

9 

36 

28 

52 

. 5430 

. 4570 

.5283 

. 6521 

.1558 

.3223 

. 4876 

. 5623 

8 

32 

32 

53 

. 5452 

. 4548 

.5278 

. 6572 

.1551 

.3227 

. 4396 

. 5604 

7 

28 

36 

54 

. 5474 

. 4526 

.5273 

. 6623 

.1544 

.3230 

. 4415 

. 5585 

6 

24 

40 

55 

.65496 

.34504 

1.5268 

.86674 

1.1537 

1.3233 

.24434 

.75566 

5 

20 

44 

56 

. 5518 

. 4482 

.5263 

. 672-3 

.1531 

.3237 

. 4453 

. 5547 

4 

16 

48 

57 

. 5540 ! 

. 4460 

.5258 

. 6775 

.1524 

.3240 

. 4472 

. 5528 

3 

12 

52 

58 

. 5562 

. 4438 

.5253 

. 6826 

.1517 

.32 i3 

. 4491 

. 5509 

2 

8 

56 

59 

. 5584 j 

. 4416 

.5248 

. 6878 

.1510 

.3247 

. 4510 

. 5490 

1 

4 

44 

60 

. 5606 

. 4394 

.5242 

. 6929 

.1504 

.3250 

. 4529 

. 5471 

0 

16 

M. S. 

M ! 

Cosine. 1 

Yrs.Sin. 

Secante. 1 

CotaugJ' 

f till Lit. I 

Cosec’ntiYrs.Cos 

Sine. 

M 

M.S. 

8 h 

130° 



Natural. 



49° 

3 h 

. 




































250 


Natural Lines. 


2 h 

41° 

Natural Trigonometrical 

Functions 

1 

33° 

9 h 

M.S. 

M 

Sine. 

Vrs.Oos. 

Cosec'nte 

Tang. 

Cotang. 

Seoante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

11 

0 

.65606 

.34394 

1.5242 

.86929 

1.1504 

1.3250 

.24529 

.75471 

60 

16 

4 

1 

. 5628 

. 4372 

.5237 

. 6980 

.1497 

.3253 

. 4548 

. 5452 

59 

56 

8 

2 

. 5650 

. 4350 

.5232 

. 7031 

.1490 

.3257 

. 4567 

. 5433 

58 

52 

12 

3 

. 5672 

. 4328 

.5227 

. 7082 

.1483 

.3260 

. 4586 

. 5414 

57 

48 

16 

4 

. 5694 

. 4306 

.5222 

. 7133 

.1477 

.3263 

. 4605 

. 5394 

56 

44 

20 

5 

.65716 

.34284 

1.5217 

.87184 

1.1470 

1.3267 

.24624 

.75375 

55 

40 

24 

6 

. 5737 

. 4262 

.5212 

. 7235 

.1463 

.3270 

. 4644 

. 5356 

54 

36 

28 

7 

. 5759 

. 4240 

.5207 

. 7287 

.1456 

.3274 

. 4663 

. 6337 

53 

32 

32 

8 

. 5781 

. 4219 

.5202 

. 7338 

.1450 

.3277 

. 4682 

. 5318 

52 

28 

36 

9 

. 5803 

. 4197 

.5197 

. 7389 

.1443 

.3280 

. 4701 

. 5299 

51 

24 

40 

10 

.65825 

.34175 

1.5192 

.87441 

1.1436 

1.3284 

.24720 

.75280 

50 

20 

44 

11 

. 5847 

. 4153 

.5187 

. 7492 

.1430 

.3287 

. 4739 

. 5261 

49 

16 

48 

12 

. 5869 

. 4131 

.5182 

. 7543 

.1423 

.3290 

. 4758 

. 5241 

48 

12 

52 

13 

. 5891 

. 4109 

.5177 

. 7595 

.1416 

.3294 

. 4778 

. 5222 

47 

8 

56 

14 

. 5913 

. 4087 

.5171 

. 7646 

.1409 

.3297 

. 4797 

. 5203 

40 

4 

45 

15 

.65934 

.34065 

1.5166 

.87698 

1.1403 

1.3301 

.24816 

.75184 

45 

15 

4 

16 

. 5956 

. 4043 

.5161 

. 7749 

.1396 

.3304 

. 4835 

. 5165 

44 

56 

8 

17 

. 5978 

. 4022 

.5156 

. 7801 

.1389 

.3307 

. 4854 

. 5140 

43 

52 

12 

18 

. 6000 

. 4000 

.5151 

. 7852 

.1383 

.3311 

. 4873 

. 5126 

42 

48 

16 

19 

. 6022 

. 3978 

.5146 

. 7904 

.1376 

.3314 

. 4893 

. 5107 

41 

44 

20 

20 

.66044 

.33956 

1.5141 

.87955 

1.1369 

1.3318 

.24912 

.75088 

40 

40 

24 

21 

. 6066 

. 3934 

.5136 

. 8007 

.1363 

.3321 

. 4931 

. 5069 

39 

36 

28 

22 

. 6087 

.'3912 

.5131 

. 8058 

.1356 

.3324 

. 4950 

. 5049 

38 

32 

32 

23 

. 6109 

. 3891 

.5126 

. 8110 

.1349 

.3328 

. 4970 

. 5030 

37 

28 

36 

24 

. 6131 

. 3869 

.5121 

. 8162 

.1343 

.3331 

. 4989 

. 5011 

36 

24 

40 

25 

.66153 

.33847 

1.5116 

.88213 

1.1336 

1.3335 

.25008 

.74992 

35 

20 

44 

26 

. 6175 

. 3825 

.5111 

. 8265 

.1329 

.3338 

. 5027 

. 4973 

34 

16 

48 

27 

. 6197 

. 3803 

.5106 

. 8317 

.1323 

.3342 

. 5047 

. 4953 

33 

12 

52 

28 

. 6218 

. 3781 

.5101 

. 8369 

.1316 

.3345 

. 5066 

. 4934 

32 

8 

56 

29 

. 6240 

. 3760 

.5096 

. 8421 

.1309 

.3348 

. 5085 

. 49 L5 

31 

4 

40 

30 

.66262 

.33738 

1.5092 

.88472 

1.1303 

1.3352 

.25104 

.74895 

30 

14 

4 

31 

. 6284 

. 3716 

.5087 

. 8524 

.1296 

.3355 

. 5124 

. 4876 

29 

56 

8 

32 

. 6305 

. 3694 

.5082 

. 8576 

.1290 

.3359 

. 5143 

. 4857 

28 

52 

12 

33 

. 6327 

. 3673 

.5077 

. 8628 

.1283 

.3362 

. 5162 

. 4838 

27 

48 

16 

34 

. 6349 

. 3651 

.5072 

. 8680 

.1276 

.3366 

. 5181 

. 4818 

26 

44 

20 

35 

.66371 

.33629 

1.5067 

.88732 

1.1270 

1.3369 

.25201 

.74799 

25 

40 

24 

36 

. 6393 

. 3607 

.5062 

. 8784 

.1263 

.3372 

. 5220 

. 4780 

24 

36 

28 

37 

. 6414 

. 3586 

.5057 

. 8836 

.1257 

.3376 

. 5239 

. 4760 

23 

32 

32 

38 

. 6436 

. 3564 

.5052 

. 8888 

.1250 

.3379 

. 5259 

. 4741 

22 

28 

36 

39 

. 6458 

. 3542 

.5047 

. 8940 

.1243 

.3383 

. 5278 

. 4722 

21 

24 

40 

40 

.66479 

.33520 

1.5042 

.88992 

1.1237 

1.3386 

.25297 

.74702 

20 

20 

44 

41 

. 6501 

. 3499 

.5037 

. 9044 

.1230 

.3390 

. 5317 

. 4683 

19 

16 

48 

42 

. 6523 

. 3477 

.5032 

. 9097 

.1224 

.3393 

. 5336 

. 4664 

18 

12 

52 

43 

. 6545 

. 3455 

.5027 

. 9149 

.1217 

3397 

. 5355 

. 4644 

17 

8 

56 

44 

. 6566 

. 3433 

.5022 

. 9201 

.1211 

3400 

. 5375 

. 4625 

16 

4 

41 

45 

.66588 

.33412 

1.5018 

.89253 

1.1204 

1.3404 

.25394 

.74606 

15 

13 

4 

46 

. 6610 

. 3390 

.5013 

. 9306 

.1197 

.3407 

. 5414 

. 4586 

14 

56 

8 

47 

. 6631 

. 3368 

.5008 

. 9358 

.1191 

.3411 

. 5433 

. 4567 

13 

52 

12 

48 

. 6653 

. 3347 

.5003 

. 9410 

.1184 

.3414 

. 5452 

. 4548 

12 

48 

16 

■ 49 

. 6675 

. 3325 

.4998 

. 9463 

.1178 

.3418 

. 5472 

4528 

11 

44 

20 

50 

.66697 

.33303 

1.4993 

.89515 

1.1171 

1.3421 

.25491 

.74509 

10 

40 

24 

51 

. 6718 

. 3282 

.4988 

. 9567 

.1165 

.3425 

. 5510 

. 4489 

9 

36 

28 

52 

. 6740 

. 3260 

.4983 

. 9620 

.1158 

.3428 

. 5530 

. 4470 

8 

32 

32 

53 

. 6762 

. 3238 

.4979 

. 9672 

.1152 

.3132 

. 5549 

. 4450 

7 

28 

36 

54 

. 6783 

. 3217 

.4974 

. 9725 

.1145 

.3435 

. 5569 

. 4431 

6 

24 

40 

55 

.66805 

.33195 

1.4969 

.89777 

1.1139 

1.3439 

.25588 

.74412 

5 

20 

44 

56 

. 6826 

. 3173 

.4964 

. 9830 

.1132 

.3442 

. 5608 

. 4392 

4 

16 

48 

57 

. 6848 

. 3152 

.4959 

. 9882 

.1126 

.3446 

. 5627 

. 4373 

3 

12 

52 

68 

. 6870 

. 3130 

.4954 

. 9935 

.1119 

.3449 

. 5647 

. 4353 

2 

8 

56. 

59 

. 6891 

. 3108 

.4949 

. 9988 

.1113 

.3453 

. 5660 

. 4334 

1 

4 

48 

60 

. 6913 

. 3087 

.4945 

.90040 

.1106 

.3456 

. 5685 

. 4314 

0 

Ifi 

M. S. 

M 

Cosine. 

Vrs.Sin. 

Seeante. 

Co£ang. 

Tangent. 

Cosec’ut 

Vrs. Cos 

Sine. 

M 

M.S. 

8 h 

131 

0 



Natural. 




00 

o 

3 h 























Natural Lines, 


251 


2 h 

42° 

Natural Trigonometrical Functions 

137° 

9 h 

M.S. 

M 

Sine. 

Yrs. Cos. 

Cosec'nte 

Tang. 

Cotaug. 

Secante. 

Yrs. Sin 

Cosine. 

M 

M.S. 

48 

0 

.66913 

.33087 

1.4945 

.90040 

1.1106 

1.3456 

.25685 

.74314 

60 

lfi 

4 

1 

. 6935 

. 3065 

.4940 

. 0093 

.1100 

.3160 

. 5705 

. 4295 

59 

56 

8 

2 

. 6956 

. 3044 

.4935 

. 0146 

.1093 

.3463 

. 5724 

. 4275 

58 

52 

12 

3 

. 6978 

. 3022 

.4930 

. 0198 

.1086 

.3167 

. 5744 

. 4266 

57 

48 

16 

4 

. 6999 

. 3000 

.4925 

. 0251 

.1080 

.3470 

. 5763 

. 4236 

56 

44 

20 

5 

.67021 

.32979 

1.4921 

.90304 

1.1074 

1.3474 

.25783 

.74217 

55 

40 

24 

6 

. 7043 

. 2957 

.4916 

. 0357 

.1067 

.3477 

. 5802 

. 4197 

54 

36 

28 

7 

. 7064 

. 2936 

.4911 

. 0410 

.1061 

.3481 

. 5822 

. 4178 

53 

32 

32 

8 

. 7086 

. 2914 

.4906 

. 0463 

.1054 

.3485 

. 5841 

. 415S 

52 

28 

36 

9 

. 7107 

. 2893 

.4901 

. 0515 

.1048 

.3488 

. 5861 

. 4139 

51 

24 

40 

10 

.67129 

.32871 

1.4897 

.90568 

1.1041 

1.3492 

.25880 

.74119 

50 

20 

44 

11 

. 7150 

. 2849 

.4892 

. 0621 

.1035 

.3495 

. 5900 

. 4100 

49 

16 

48 

12 

. 7172 

. 2828 

.4887 

. 0674 

.1028 

.3499 

. 5919 

. 4080 

48 

12 

52 

13 

. 7194 

. 2806 

.4882 

. 0727 

.1022 

.3502 

. 5939 

. 4061 

47 

8 

56 

14 

. 7215 

. 2785 

.4877 

. 0780 

.1015 

.3506 

. 5959 

. 4041 

46 

4 

49 

15 

.67237 

.32763 

1.4S73 

.90834 

1.1009 

1.3509 

.25978 

.74022 

45 

11 

4 

16 

. 7258 

. 2742 

.4808 

. 0887 

.1003 

.3513 

. 5998 

. 4002 

44 

56 

8 

17 

. 7280 

. 2720 

.4863 

. 0940 

.0996 

.3517 

. 6017 

. 3983 

43 

52 

12 

IS 

. 7301 

. 2699 

.4858 

. 0993 

.0990 

.3520 

. 6037 

. 3963 

42 

48 

16 

19 

. 7323 

. 2677 

.4854 

• 1046 

.0983 

.3524 

. 6056 

. 3943 

41 

44 

20 

20 

.67344 

.32656 

1.4849 

.91099 

1.0977 

1.3527 

.26076 

.73924 

40 

40 

24 

21 

. 7366 

. 2634 

.4844 

. 1153 

.0971 

.3531 

. 6096 

. 3904 

39 

36 

28 

22 

. 7387 

. 2613 

.4S39 

• 1206 

.0964 

.3534 

. 6115 

. 3885 

38 

32 

32 

23 

. 7409 

. 2591 

.4835 

. 1259 

.0958 

.3538 

. 6135 

. 3865 

37 

28 

36 

24 

. 7430 

. 2570 

.4830 

. 1312 

.0951 

.3542 

. 6154 

. 3845 

36 

24 

40 

25 

.67452 

.32548 

1.4825 

.91366 

1.0945 

1.3515 

.26174 

.73826 

35 

20 

44 

26 

. 7473 

. 2527 

.4821 

. 1419 

.0939 

.3549 

. 6194 

. 3806 

34 

16 

48 

27 

. 7495 

. 2505 

.4816 

. 1473 

.0932 

.3552 

. 6213 

. 3787 

33 

12 

52 

28 

. 7516 

. 2484 

.4811 

. 1526 

.0926 

.3556 

. 6233 

. 3767 

82 

8 

66 

29 

. 7537 

. 2462 

.4806 

. 1580 

.0919 

.3560 

. 6253 

. 3747 

31 

4 

50 

30 

.67559 

.32441 

1.4802 

•91633 

1.0913 

1.3563 

.26272 

.73728 

30 

10 

4 

31 

. 7580 

. 2419 

.4797 

. 1687 

.0907 

.3507 

. 6292 

. 37uS 

29 

56 

8 

32 

. 7602 

. 2398 

.4792 

• 1740 

.0900 

.3571 

. 6311 

. 3688 

28 

52 

12 

33 

. 7623 

. 2377 

.4788 

• 1794 

.0894 

.3574 

. 6331 

. 3669 

27 

48 

16 

3^ 

• 7645 

. 2355 

.4783 

. 1847 

.0888 

.3578 

. 6351 

. 3649 

26 

44 

20 

35 

.67666 

.32334 

1.4778 

.91901 

1.0881 

1.3581 

.26371 

.73629 

25 

40 

24 

36 

. 7688 

. 2312 

.4774 

. 1955 

.0875 

.3585 

. 6390 

. 3610 

24 

36 

, 28 

37 

. 7709 

. 2291 

.4769 

• 2008 

.0868 

.3589 

6410 

. 3590 

23 

32 

32 

38 

. 7730 

. 2269 

.4764 

• 2002 

.0862 

.3592 

. 6430 

. 3570 

22 

28 

36 

39 

• 7752 

. 2248 

.4760 

. 2116 

.0856 

.3596 

. 6449 

. 3551 

21 

24 

40 

40 

.67773 

.32227 

1.4755 

•92170 

1.0849 

1.3600 

.26469 

.73531 

20 

20 

44 

41 

• 7794 

. 2205 

.4750 

. 2223 

.0843 

.3603 

. 6489 

. 3511 

19 

16 

48 

42 

. 7816 

. 2184 

.4746 

• 2277 

.0837 

.3607 

. 6508 

. 3491 

IS 

12 

52 

43 

. 7837 

. 2163 

.4741 

. 2331 

.0830 

.3611 

. 6528 

. 3472 

17 

8 

56 

44 

. 7859 

. 2141 

.4736 

. 2385 

.0824 

.3614 

. 6548 

. 3452 

16 

4 

51 

45 

.67880 

.32120 

1.4732 

.92439 

1.0818 

1.3618 

.26568 

.73432 

15 

9 

4 

46 

• 7901 

. 2098 

.4727 

. 2493 

.0812 

;3622 

. 6587 

. 3412 

14 

56 

8 

47 

. 7923 

. 2077 

.4723 

. 2547 

.0805 

.3625 

. 6607 

. 3393 

13 

52 

12 

48 

. 7944 

. 2056 

.4718 

. 2601 

.0799 

.3629 

. 6627 

. 3373 

12 

48 

16 

49 

. 7965 

. 2034 

.4713 

. 2655 

.0793 

.3633 

. 6647 

. 3353 

11 

44 

20 

50 

.67987 

.32013 

1.4709 

.92709 

1.0786 

1.3636 

.26666 

.73333 

10 

40 

24 

51 

. 8008 

. 1992 

.4704 

. 2763 

.0780 

.3640 

. 6686 

. 3314 

9. 

36 

28 

52 

. 8029 

. 1970 

.4699 

. 2817 

.0774 

.3044 

. 6706 

. 3294 

8 

32 

32 

53 

. 8051 

. 1949 

.4695 

. 2871 

.0767 

.3647 

. 6726 

. 3,274 

7 

28 

36 

54 

. 8072 

. 1923 

.4690 

. 2926 

.0701 

.3651 

. 6746 

. 3254 

6 

24 

40 

65 

.68093 

.31907 

14686 

.92980 

1.0755 

1.3655 

.26765 

.73231 

5 

20 

44 

56 

. 8115 

. 1885 

.4681 

. 3034 

.0749 

.3658 

. 6785 

. 3215 

4 

16 

48 

57 

. 8136 

. 1864 

.4676 

. 3088 

.0742 

.3662 

. 6805 

. 3195 

3 

12 

52 

58 

. 8157 

. 1843 

.4672 

. 3143 

.0736 

.3666 

. 6825 

. 3175 

2 

8 

56 

59 

. 8178 

. 1821 

.4667 

. 3197 

.0730 

.3669 

. 6845 

. 3155 

1 

4 

53 

60 

. S200 

. 1800 

.4663 

. 3251 

.0724 

.3673 

. 6865 

. 3135 

0 

8 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secante. 

Cotaug. 

Tangent. 

Cosec’nt. 

Yrs. Cos 

Sine. 

SI 

M.S 

8“ 

132 c 




Natural. 



47° 

3 L 




































m 


Natural, Lines. 



43° 

Natural Trigonometrical Functions. 

136° 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

jCosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

5i 

0 

.68200 

.31800 

1.4663 

.93251 

1.0724 

1.3673 

.26865 

.73135 

60 

8 

4 

1 

. 8221 

. 1779 

.4658 

. 3306 

.0717 

.3677 

. 6884 

. 3115 

59 

56 

8 

2 

. 8242 

. 1758 

.4654 

. 3360 

.0711 

.3681 

. 6904 

. 3096 

58 

52 

12 

3 

. 8264 

. 1736 

.4649 

. 3415 

.0705 

.3 84 

. 6924 

. 3076 

57 

48 

16 

4 

. 8285 

. 1715 

.4614 

. 3469 

.0699 

.3688 

. 6944 

. 3056 

56 

44 

20 

5 

.68306 

.31694 

1.4640 

.93524 

1.0692 

1.3692 

.26964 

.73036 

55 

40 

24 

6 

. 8327 

. 1673 

.4635 

. 3578 

.0686 

.3695 

. 6984 

. 3016 

54 

36 

28 

7 

. 8349 

. 1651 

.4631 

. 3633 

.0680 

.3699 

. 7004 

. 2996 

53 

32 

32 

8 

. 8370 

. 1630 

.4626 

. 3687 

.0674 

.3703 

. 7023 

. 2976 

52 

28 

36 

9 

. 8391 

. 1609 

.4622 

. 3742 

.0667 

.3707 

. 7043 

. 2956 

51 

24 

40 

10 

.68412 

.31588 

1.4617 

.93797 

1.0661 

1.3710 

.27063 

.72937 

50 

20 

44 

11 

. 8433 

. 1566 

.4613 

. 3851 

.0655 

.3714 

. 7083 

. 2917 

49 

16 

48 

12 

. 8455 

. 1545 

.4608 

. 691)6 

.0649 

.3718 

. 7103 

. 2897 

48 

12 

62 

13 

. 8476 

. 1624 

.4604 

. 3961 

.0643 

.3722 

. 7123 

. 2877 

47 

8 

66 

14 

. 8497 

. 1503 

.4599 

. 4016 

.0636 

.3725 

. 7143 

. 2857 

46 

4 

53 

15 

.68518 

.314S2 

1.4595 

.94071 

1.0630 

1.3729 

.27163 

.72837 

45 

7 

4 

16 

. 8539 

. 1460 

.4590 

. 4125 

.0624 

.3733 

. 7183 

. 2817 

44 

56 

8 

17 

. 8561 

. 1439 

.4586 

. 4180 

.0618 

.3737 

. 7203 

. 2797 

43 

52 

12 

18 

. 8582 

. 1418 

.4581 

. 4235 

.0612 

.3740 

. 7223 

. 2777 

42 

48 

16 

19 

. 8603 

. 1397 

.4577 

. 4290 

.0605 

.3744 

. 7243 

. 2757 

41 

44 

20 

20 

.68624 

.31376 

1.4572 

.94345 

1.0599 

1.3748 

.27263 

.72737 

40 

40 

24 

21 

. 8645 

. 1355 

.4568 

. 4400 

.0593 

.3752 

. 7283 

. 2717 

39 

36 

28 

22 

. 8666 

. 1333 

.4563 

. 4455 

.0587 

.3756 

. 7302 

. 2697 

38 

32 

32 

23 

. 8688 

. 1312 

.4559 

. 4510 

.0581 

.3759 

. 7322 

. 2677 

37 

28 

36 

24 

. 8709 

. 1291 

.4554 

. 4565 

.0575 

.3763 

. 7342 

. 2657 

36 

24 

40 

25 

.68730 

.31270 

1.4550 

.94620 

1.0568 

1.3767 

.27362 

.72637 

35 

20 

44 

26 

. 8751 

. 1249 

.4545 

. 4675 

.0562 

.3771 

. 7382 

. 2617 

34 

16 

48 

27 

. 8772 

. 1228 

.4541 

. 4731 

.0556 

.3774 

. 7402 

. 2597 

33 

12 

52 

28 

. 8793 

. 1207 

.4536 

. 4786 

.0550 

.3778 

. 7422 

. 2577 

32 

8 

56 

29 

. 8814 

. 1186 

.4532 

. 4S41 

.0544 

.3782 

. 7442 

. 2557 

31 

4 

54 

30 

.68835 

.31164 

1.4527 

.94896 

1.0538 

1.3786 

.27462 

.72537 

30 

6 

4 

31 

. 8856 

. 1143 

.4523 

. 4952 

.0532 

.3790 

. 7482 

. 2517 

29 

56 

8 

32 

. 8878 

. 1122 

.4518 

. 5007 

.0525 

.3794 

. 7503 

. 2497 

28 

52 

12 

33 

. 8899 

. 1101 

.4514 

. 5062 

.0519 

.3797 

. 7523 

. 2477 

27 

48 

16 

34 

. 5920 

. 1080 

.4510 

. 5118 

.0513 

.3801 

.. 7543 

. 2457 

26 

44 

20 

35 

.68941 

.31059 

1.4505 

.95173 

1.0507 

1.3805 

.27563 

.72437 

25 

40 

24 

36 

. 8962 

. 1038 

.4501 

. 5229 

.0501 

.3809 

. 7583 

. 2417 

24 

36 

28 

37 

. 8983 

. 1017 

.4496 

. 5284 

.0195 

.3813 

. 7603 

. 2397 

23 

32 

32 

38 

. 9004 

. 0996 

.4492 

. 5340 

.0489 

.3816 

. 7623 

. 2377 

22 

28 

36 

39 

. 9025 

. 0975 

.4487 

. 5395 

.0483 

.3820 

. 7643 

. 2357 

21 

24 

40 

40 

.69046 

.30954 

1.4483 

.95451 

1.0476 

1.3824 

.27663 

.72337 

20 

20 

44 

41 

. 9067 

. 0933 

.4479 

. 5506 

.0470 

.3^28 

. 7683 

. 2317 

19 

16 

48 

42 

. 9088 

. 0912 

.4474 

. 5562 

.0464 

.3832 

. 7703 

. 2it97 

18 

12 

52 

43 

. 9109 

. 0891 

.4470 

. 5618 

.0458 

.3886 

. 7723 

. 2277 

17 

8 

56 

44 

. 9130 

. 0870 

.4465 

. 5673 

.0452 

.3839 

. 7743 

. 2256 

16 

4 

55 

45 

.69151 

.30849 

1.4461 

.95729 

1.0446 

1.8843 

.27764 

.72236 

15 

5 

4 

46 

. 9172 

. 082S 

.4457 

. 57S5 

.0440 

.3847 

. 7784 

. 2216 

14 

56 

8 

47 

. 9193 

. 0807 

.4452 

. 5841 

.0434 

.3851 

. 7804 

. 2196 

13 

52 

12 

48 

. 9214 

. 0786 

.4448 

. 5896 

.0428 

.3855 

. 7824 

. 2176 

12 

48 

• 16 

49 

. 9235 

. 0765 

.4443 

. 5952 

.0422 

.3859 

. 7844 

. 2156 

11 

44 

20 

50 

.69256 

.30744 

1.4439 

.96008 

1.0416 

1.3863 

.27864 

.72136 

10 

40 

24 

51 

. 9277 

. 0723 

.4435 

. 6064 

.0410 

.3867 

. 7884 

. 2115 

9 

36 

28 

'52 

. 9298 

. 0702 

.4430 

. 6120 

.0404 

.3870 

. 7901 

. 2095 

8 

32 

32 

53 

. 9319 

. 0684 

.4426 

. 6176 

.0397 

.3874 

. 7925 

. 2075 

7 

28 

36 

54 

. 9340 

. 0660 

.4422 

. 6232 

.0391 

.3878 

. 7945 

. 2055 

6 

21 

40 

55 

.69361 

.30639 

1.4417 

.96288 

1.0385 

1.3882 

.27965 

.72035 

5 

20 

44 

56 

. 9382 

. 0618 

.4413 

. 6344 

.0379 

.3886 

. 7985 

. 2015 

4 

16 

48 

57 

. 9403 

. 0597 

.4408 

. 6400 

.0373 

.3890 

. 8005 

. 1994 

3 

12 

52 

58 

. 9424 

. 0576 

.4404 

. 6456 

.0367 

.3894 

. 8026 

. 1974 

2 

8 

56 

59 

. 9445 

. 0555 

.4400 

. 6513 

.0361 

.3898 

. 8046 

. 1954 

1 

4 

5G 

60 

. 9466 

. 0534 

.4395 

. 6569 

.0355 

.3902 

. 8066 

. 1934 

0 

•£ 

M. S. 

M 

Cosine. 

Yrs.Siu, 

Secaute. 

Cotaug., 

Tangent. 

Cosec’nt! Vrs.Cos 

Sine. 

M 

M.S. 

s h 

133' 

3 



Natural. 




46° 

3 h 

































Natural Lines. 


253 


2 h 

44 c 


Natural Trigonometrical Functions 

135° 

9 h 

M.S. 

ii 

Sine. 

Yrs. Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secauie. 

Vrs.Sin 

Cosi ne. 

M 

M.S. 

56 

0 

.69466 

.30531 

1.4395 

.96563 

1.0355 

1 3902 

.28066 

.71934 

60 

4 

4 

l 

. 9487 

. 0513 

.4391 

. 6625 

.0349 

.3905 

. 8086 

1914 

59 

55 

8 

2 

. 9508 

. 0492 

.4387 

. 6681 

.0843 

.3 )09 

. 8106 

. 1893 

58 

52 

12 

3 

. 9528 

. 0171 

.4382 

. 6738 

.0337 

.3913 

. 8127 

. 1873 

57 

48 

1G 

4 

. 9519 

. 0470 

.4378 

. 6794 

.0331 

.8947 

. 8147 

. 1853 

56 

44 

20 

5 

-69570 

.30430 

1.4374 

.96850 

1.0325 

1.392 L 

.28167 

.71833 

55 

40 

24 

G 

. 9591 

. 0409 

.4370 

. 6.)07 

.0319 

.3925 

. 8187 

. 1813 

54 

36 

28 

7 

. 9612 

. 0388 

.4365 

. 6963 

.0313 

.3929 

. 8208 

. 1792 

53 

32 

32 

8 

. 91 33 

. 0367 

.4361 

. 7020 

.0307 

.393 1 

. 8228 

. 1772 

52 

28 

36 

9 

. 9651 

- 0346 

.4367 

. 7076 

.0-01 

.3937 

. 8248 

. 1752 

51 

24 

40 

10 

.69675 

.30325 

1.1352 

.97133 

1.0-95 

1.3941 

.28268 

.71732 

50 

20 

44 

11 

. 9696 

. 0304 

.4348 

. 7189 

.0289 

.3945 

. 8289 

. 1711 

49 

16 

4S 

12 

. 9716 

. 0283 

.4344 

. 7246 

.02*3 

.394) 

. 8309 

. 1691 

48 

12 

52 

13 

• 9737 

. 0263 

.4339 

. 7302 

.0277 

.3953 

. 8329 

. 1671 

47 

8 

66 

14 

. 9758 

. 0242 

.4335 

. 7359 

.0 271 

.3957 

. 8349 

. 1670 

46 

4 

67 

15 

.69779 

.30221 

1.4331 

•97416 

1.0265 

1.396 ) 

.28370 

.71630 

45 

3 

4 

16 

. 98()0 

. 0200 

.4327 

• 7472 

.0259 

.8964 

. 8890 

. 1610 

44 

56 

8 

17 

. 9821 

. 0179 

.4322 

. 7529 

.0253 

.3968 

. 8410 

. 15-9 

43 

52 

12 

IS 

. 9841 

. 015S 

.4318 

• 7586 

.0247 

.3972 

. 8431 

. 17 69 

42 

48 

10 

19 

. 9862 

. 0138' 

-4314 

• 7643 

.0241 

.397 3 

. 8451 

. 1549 

41 

41 

20 

20 

•69583 

.30117 

1-4310 

•97699 

1.0235 

1.3980 

.28471 

.71529 

40 

40 

24 

21 

• 9904 

. 0096 

-4305 

• 7756 

.0229 

.39 4 

. 8 92 

. 1508 

39 

36 

28 

22 

• 9925 

. 0075 

.4301 

. 7813 

.0223 

.898 4 

. 8512 

. 1488 

38 

32 

32 

23 

. 9945 

. 0054 

.4297 

. 7870 

.0218 

.3992 

. 8532 

. 1468 

37 

28 

3G 

24 

• 9966 

. 0034 

.4292 

• 7927 

.0212 

.3o 9 6 

. 8; 53 

. 1447 

36 

24 

40 

25 

-69987 

.30013 

1.4288 

•97984 

1.0206 

’ 1.4000 

.28573 

.71427 

35 

20 

41 

26 

-70008 

.29992 

-4284 

• 8041 

.0200 

.4004 

. 8593 

. 1406 

34 

16 

48 

27 

• 0029 

. 9971 

.4280 

• 8098 

.0194 

.4008 

. 8614 

. 1386 

33 

12 

52 

28 

- 0049 

. 9950 

-4276 

• 8155 

.0188 

.4012 

. 8i 34 

. 1366 

82 

8 

5G 

29 

• 0070 

. 9930 

.4271 

• 8212 

.0182 

.4016 

. 8654 

. 1345 

31 

4 

58 

30 

-70091 

.29. 09 

1.4267 

•98270 

1.0176 

1.4020 

.28675 

.71325 

30 

2 

4 

31 

• 0112 

. 9888 

•4263 

• 8327 

.0170 

.4024 

. 8695 

. 1305 

29 

56 

8 

32 

- 0132 

. 9867 

4259 

. 8384 

.0164 

.402S 

. S716 

. 1284 

28 

52 

12 

33 

• 0153 

. 9847 

.4254 

. 8441 

.0158 

.4032 

. 8736 

. 1-64 

27 

48 

16 

34 

• 0174 

. 9826 

.4250 

• S499 

-0152 

.4036 

. 8756 

. 1243 

26 

41 

20 

35 

-70194 

.29805 

1.4246 

•98556 

1.0146 

1.4040 

.28777 

.71223 

‘25 

40 

24 

36 

- 0215 

. 9785 

.4242 

• 8613 

.0141 

.4044 

. 8797 

. 1203 

24 

36 

28 

37 

. 0236 

. 9764 

.4238 

• 8671 

.0135 

.4048 

8818 

. 1182 

23 

32 

32 

38 

- 0257 

. 9743 

.4233 

. 872s 

.0129 

.4052 

. 8 38 

. 1162 

22 

28 

36 

39 

- 0277 

. 9722 

.4229 

• 8 786 

.0123 

1.4056 

. 8859 

. 1141 

21 

24 

40 

40 

-70298 

.29702 

1.4225 

•98843 

1.0117 

.4060 

.28879 

.71121 

20 

20 

44 

41 

• 0319 

. 9681 

.4221 

• 8901 

.0111 

.4065 

. 8899 

. 1100 

19 

16 

48 

42 

. 0339 

. 9660 

.4217 

• 8958 

.0105 

.4069 

. 8920 

. 108(1 

18 

12 

52 

43 

. 0360 

. 9640 

.4212 

• 9016 

.00 9 

.4073 

. 8940 

. 1059 

17 

8 

56 

41 

. 0381 

. 9619 

.4208 

. 9073 

.0093 

.4077 

. 8961 

. 1039 

16 

4 

59 

45 

•70401 

.29598 

1.4204 

■99131 

1.0088 

1.4081 

.28981 

.7101S 

15 

1 

4 

46 

. 0422 

. 9578 

.4200 

• 9189 

.0082 

.4085 

. 9002 

. 0998 

14 

56 

S 

47 

• 0143 

. 9557 

.4196 

. 9216 

.0076 

.4089 

. 9022 

. 0977 

13 

52 

12 

48 

. 0463 

. 9536 

.4192 

. 9304 

.0070 

.4093 

. 9043 

. 0957 

12 

48 

1G 

49 

- 0484 

. 9516 

.4188 

. 9362 

.0064 

.4097 

. 9063 

. 0938 

11 

44 

20 

50 

.705U5 

.29495 

1.4183 

.99420 

1.0058 

1.4101 

.29084 

.70916 

10 

40 

24 

51 

. 0525 

. 9475 

.4179 

. 9478 

.0052 

.4105 

. 9104 

. 0805 

9 

36 

28 

52 

. 0546 

. 9454 

.4175 

. 95: 0 

.0047 

.4109 

. 9125 

. OS75 

8 

32 

32 

53 

. 0566 

. 9433 

.4171 

• 9693 

.0041 

.4113 

. 9145 

. 0854 

7 

2S 

3C 

54 

. 0587 

. 9413 

.4167 

. 9651 

.0035 

.4117 

. 9166 

. 08: 4 

6 

24 

40 

65 

.70608 

.29392 

1.4163 

.99709 

1.0029 

1.4122 

.29186 

.70813 

5 

20 

41 

50 

. 0628 

. 9372 

.4159 

. 9707 

.0023 

.4126 

. 9207 

. 0703 

4 

16 

4s 

57 

. 0649 

. 9351 

.4154 

. 9826 

.< 017 

.4130 

. 9228 

. 0772 

3 

12 

52 

58 

. 0669 

. 9330 

.4150 

. 9884 

.0012 

.4184 

. 9. 4S 

. 0752 

2 

8 

56 

59 

. 0690 

. 9310 

.4146 

. 9942 

.0006 

.4138 

. 9269 

. 0731 

1 

4 

60 

60 

. 0711 

. 9289 

.4142 

1.0000 

1.0000 

.4142 

. 9289 

• 0711 

0 

O 

M . S. 

M 

Cosine. 

Yrs. Siu. 

Secante. 

Cotang. 

tangent. 

Cosec’nt.iVrs.Cos 

Sine. | 

M 

AI S. 

8 » 

134° 



Mai ural. 




45° 

3 h | 




































Mechanics.—Statics. 




MECHANICS. 

Mcclianics is that branch of Natural Philosophy -which treats of the action 
of force, motion, and power. Mechanics is divided into four parts, namely, 
Statics the science of forces in equilibrium. 

Dynamics , the science of forces in motion, it produces power or effect. 
Hydrostatics, the science of fluids in equilibrium 

hydrodynamics the science of fluids in motion, its causes, power or effects. 

Statics* Lever* Momentum* 


Lever is an inflexible bar, supported in one point called the Fulcrum, or, 
centre of motion. The length of a lever is measured from the fulcrum to where 
the force or resistance acts, (when the force acts at right angles to the lever) or, 
the length of a lever is measured from the fulcrum at right angles to the direc¬ 
tion of the force. 

W— Weight, and l — lever for TF) Q x ,. 1or , 

F = Force, and Z = lever for F\ See Fl S* 139 * 

Momentum is the product of force or weight, multiplied by the length of 
the lever it acts upon. 

The products Wl and FL are called Statics Momentums ; when these mo- 
men turns are equal there will be no motion, and the weight TPwill balance 
the force F. When one momentum is greater than the other, there null be a 
motion, and the velocity of that motion is measured by the difference of the 
momentums. 

Levers are of three distinct hinds, with reference to the relative positions of 
the Force F, Weight W, and Fulcrum C. 

1st. Fulcrum C, is between the force F, and the weight W. 

2d. Weight W, is between the fulcrum C, and the force F. 

3d. Force F, is between the f ulcrum G, and the weight W. 

Examjde 1. Figure 139. The weight W = 68 pounds, the lever l ~ 3*86 feet 
andZ= 10 feet 6 inches. 

Required the force F= ? 


Formula 1. 


„ Wl 
f =-l 


68X3-S6 

10-6 


= 25 pounds nearly. 


a = distance between the force F and the weight W. 

The formula 3, 4, 7, 8, 11, 12, are for finding the fulcrum G, when the force 
F, weight W. and the distance a, are given. 

Example 2. Fig. 140. The force F= 360 pounds, W — 1870, and a = 8 feet, 4 
inches. 

Required the position of the fulcrum c? 


Formula 7. 


I = 


Fa 

W—F 


360X8*333 _ 2999-988 
1870 —"360 — 1510 


= 19'86 feet. 


L— 8-333-|-19-86 = 28-193 feet, the answer. 

Example 3. Fig. 144.The weight of the lever is Q — 18 pounds. The centre of 
gravity is x — 2-25 feet from the fulcrum. W = 299 pounds, l = 5-5 feet, and 


/>= 11-95. 

Required the force F = ? in pounds. 



299X5-5 — 18X2-25 
11-95 


= 134-25 pounds. 


Inclined Plane. 

Example 4. Fig. 163. A load IF — 3466 poitnds, is to be drawn up an inclined 
plane. I — 638 feet long, and 7t == 86 feet high. 

What force is required to keep the load on the inclined plane ? 

h W 86X3466 

F = ~i~ — 638 = 467-2 pounds. 





















Mbchamcs.—STATICS. 


25.1 


Example 4. Fig. 167. A Cylinder of east iron, weighing IF = 5245 pounds, is to 
be rolled up an inclined plane; the angles v — 18° 20' and v' = 8° 10' 

What force is required to keep the cylinder on the plane? 

F= W. sin.O+n - ) =5245X«n.(18 0 20'+8° 10') = 2340 pounds. 

Example 5. Fig. 168. An iron ball which weighs 398 pounds, is tied to an in¬ 
clined plane with a rope; the angle of the rope and the inclined plane is 
t/ = 16° 40', and v — 14° 30'. What force is acting on the rope ? 

„ IFsin.u 398Xsin.14 0 30' , 

F = — =-■ -.-o T -> — = 104 pounds. 

cos.n' cos.16° 40' 

Example 6. Fig. 153. What force F is required to raise a weight IF = 8469 
pounds, by a double moveable pulley ? 

F = i>F= jX 84C9 = 2117-25 pounds. 

Example 7. Fig. 156. Ilow much weight can a force F = 269 pounds lift hv 
three compound moveable pulleys ? 

IF = 2 U F = 2 1< X269 = 2152 pounds, the answer. 

Screw. 

Example 8. Fig. 172. What force is required to lift a weight IF = 16785 pounds, 
by a screw, with a pitch P — 0-125 feet, the lever being r = 5 feet. 4 inches ? 


WP 


16785X0-125 


= 62-62 pounds, the answex*. 


2*-r 2X3-14X5-333 

Including friction the force F will be 

V W{P + fd n ) 

2 nr ' 

Find the friction f on page 267. d diameter of the screw in feet. 


Wedge. 

Example 9. Fig. 169. The head of the wedge a 
l -= 16A inchesthe resistance to be separated is R = 
the force F = ? (Friction omitted.) 


= 3 inches, and length 
: 4846 pounds. Kequirod 


F = 


4846X3 


= 881 pounds. 


F - B [? +4 2 +-D] 


146 


16-5 

Including friction the force F will be, 

a 

l 

in which the friction f is to be found on page 267. 

Cateuaria. 

Example 10. An iron chain 256 feet long, weighing 1560 pounds, is to be sus¬ 
pended between two points in the same horizontal line, but 196 feet apart. 

Ilow deep will the chain bang under the line of suspension, and with what 
force will the chain act at the points of suspension ? 

Fiqare and Formula 161. we have given, 

W= iX!560 = 780 poun ds, l = £X256 = 128 feet, and a = £X196 = 98 feet. 
h =* 0-6525j/128 a — 98* = 53-73 feet, the required depth under the horizontal 
line. 

cot.n = = 1-096, or v — 4-1° 44', and 2v = 89° 28'. 

The required force will be, 

F= 


7S0X c -iu.44° 44' 

■dn.89° 28' 


= 649 pounds. 

























256 


Lever \np Static Moment "m 





























































Lever and Static Momentum. 


257 



145. F:W=r : 12. FR= Wr, , 

^ W, fi ^Wr 

F ’ F ’ 

W== ^ rmm *F 

KK r 1 r W 


146 

„ Wrr' T __ FFF' 

F - ~~TT~Tp > - 7 -’ 

RR r r 

Ti = number of revolutions of the wheels, 

n : in' = r' : R, v : v' — rr ': RR', 

V = velocity of W, v' = velocity of F. 


14 i; Wrr'r" w F R R'R" 

m& k 

R R'R" 7 rr'r" 1 

n : n" — r'r" : F F', v : v = rr'r" : 
RR'R". 

r r'r" &C. = radii of the pinions. 

F R'R"& C. = radii of the wheels. 

s'\P 

v 7 ^ 

148. 

Let P and Q represent the magnitudes and direc¬ 
tions of two forces which act to move the body B. 
By completing the parallelogram, there will be ob¬ 
tained a diagonal force F, whose magnitude and di¬ 
rection is equal to the sum P and Q. F is called 
the resultant of P and Q. 

t~ —~ - (~k'3) 

\ V^L/ 

\Jj^LSj5: 

i 

149. 

«* 

If three or more forces act in different directions 
to move a body B , find the resultant of auy two of 
them, and consider it as a single force. Between 
this and the next force find a second resultant, 
thus: P. Q , and F are magnitudes and directions of 
the forces. P+ Q = r, r-\-R = P= P-f Q+R, or P 
is the magnitude and direction of the three forces, 
P, Q, and R: 

A .* 

/ \ 

^ a/ 

\ 

L_ 

'150. 

A force Q acting (alone) on the body B, can move 
it to a in a unit of time, another force P is able to 
move it to b in the same time; now if the two 
forces act at the same time, they will move the 
| body to c. c is the resultant of a and b. 










































Pulleys. 


258 















































Funicular and Catenarian. 





















































In ..'lined Plane. 


•’TO 


sdT £ 

/ i 

/ vl/ 

A 

A 

163. f= W J 1 = Wbw.v, 

W= Fl F t 

A ~ sin.i> 

Wl 

W = —- = W COS.V. 

0 



164. 

F = W sin.(iH-i/), 

F 

tv _ 

sin.(v+vi)’ 

IF = IF cos.^+u'). 



165. w . 

^ IF sm.D 

COS.P ’ 

TF -FCOS.P' 
sin.v * 

w = IF (cos.iH-sin.v. tanr'). 

\ \ \\i 

L 

166. 

To solve an Inclined Plane by diagrams. 

F = magnitude and direction of the 
force, which is obtained by completing 
the parallelogram. 

* "k$ & 

“ By calculation see Formula, Fig. 163. 

( V7 irvf' 

V /'/i\UT/k 

< x'P ^ ? 


167. 

W = weight of the body, and direc¬ 
tion of the force of gravity ; to be drawn 
at right-angles to the base b, and F par¬ 
allel to F. 

By calculation see Formula, Fig. 164. 

/AV ^-"X T7» 

^tjT 

JWf" 


168. 

w = the force with which the body 
presses against the plane, to be drawn 
at right-angles to the plane l; then the 
parallelogram is completed. 

By calculation see Formula, Fig. 165. 

i 

1 


































Wedge and Screws 


9GI 



Wedge. 


R a 


R = 


FI 


l 7 a 

F = force acquired to drive the wedge. 


Let the line F represent the magnitude and di¬ 
rection of a force acting to move the body B on the 
line CD; then the line a represents a part of F 
which presses the body B against CD, and the line 
b represents the magnitude of the force which 
actually moves the body B. 

b = \T P — a 2 , b = F coa.v. 


h : b — sin.u : cos.v = tan.v. 
F'~F. 


Wh 

—= W tan.u. 
b 


tan.u 


F cot.v. 


172. Force by a Screw. 

P — Pitch of the screw, 
r = radius on which the force F acts. 

F : W = P : 2rc r. 


WP 

%Ttr 




F2nr 


173. Force by Compound Screws. 

P — Pitch of the large screw, 
p = Pitch of the endless screw. 

R = radius of spur-wheel for the endless 
screw. 

W: F = 4t7 2 R r : P p. 


WP r 
Att* R r’ 




FVnRr 

Pp ' 


On the spur-wheel is a cylinder by which 
the weight IV' is wound up, the formula will 
— radius of the cylinder, and 

F: W =p r’ :2nRr. 

F= w P r w _na*T. 

2 7tRr P r 






































































262 


Dynamics. 


Dynamics. 

Quantity is that which can he increased and diminished by homogeneous 
parts. Quantity is of two kinds, —geometrical and physical. 

Element is that which cannot be dissolved into two or more different 
quantities. 

Function is the product of two or more different elements. 

Force, motion, and time are simple physical elements; and 

Power, space, and work are functions or products of those elements. 

Force is a mutual tendency of bodies to attract or repel each other. Its 
physical constitution is not yet known. We only know its action on matter, 
which is recognized as pressure and measured by weight. 

Force is the first element of power and work. It is here denoted by the letter 
.Fin pounds avoirdupois. 

Motion is a continuous change of position, recognized as velocity. It is 
the second element of power and work, and here denoted by V in ft. per second. 

Time implies a continuous action recognized as duration. It is the third 
element of work, and here denoted by T in seconds. 

Power is a function of the two first elements, force Fand velocity V, or the 
product obtained by multiplying together the force F and velocity V, denoted 
P — FV. Power Fis expressed in footpounds, and called dynamic effect, of 
which there are 550 in a horse-power; or, if the velocity is measured in feet 
per minute, there will be 33,000 footpounds in a horse-power. 

Space is a function of the second and third elements, velocity Fand time 
T, denoted by S — V T, which means that the space S is the product of the 
velocity V and time T, expressed in linear feet. 

Work is a function of the three elements force F, velocity V, and time T, 
denoted by K = F V T, which means that the work K is the product obtained 
by multiplying together the three elements F, V, and T. Work may also be de¬ 
noted by K = FS, where it appears as if the work was independent of time; 
but the time is included in the space S = V T. 

Work is also denoted by K — P T, or the product of the power P and the 
time T, where it appears as if work was independent of force and velocity, 
which latter are included in the power P. Either of the three cases expresses 
the work K in units of one pound lifted one foot, called footpounds. For large 
quantities of work, this unit is too small. The work done by one man in a day 
may amount to some two millions of footpounds. It is therefore proposed to 
adopt an additional unit for work, namely, the work a horse can perform in the 
time of one hour, which is equivalent to that of eleven men working one hour 
or of one man working eleven hours, to be called a Workmanday, and 
denoted by the letter k. This unit is to express great amounts of work, such 
as the building of a house, bridge, canal, ship, large fly-wheels, and to express 
the . magnitude of steam-boiler and gunpowder explosions; also the capacity of 
heavy ordnance, to be noted in workmandays. 

One horse-power = 550 effects = 11 men’s power. 

One man’s power = 50 effects = 0-0909 horse-power. 

Example 1. What force F= ? is required to generate a power of IT = 54 
horses, with a velocity V= 15 feet per second? Find the formula for force 
which contains the given quantities H and V, which is, 


F— 


550 H 550 X 54 

v ~ 13 : 


1980 pounds, the answer. 


Example 2.—The pull of a belt over a belt-pulley is F = 350 pounds, the ra¬ 
dius of the pulley is r = 1-5 feet, making n = 52 revolutions per minute. Re¬ 
quired the horse-power H = ? transmitted by the belt. 

See formulas for circular motion. 


H = 


F r n 350 X 1-5 X 52 


= 5-2 horses. 


F 


5250 5250 

Example 3.—How many workmandays k = ? are required to raise a load of 
' — 72968000 pounds, a height of S = 25 feet. 

F S 72968000 X 25 


k = 


1980000 


1980000 


= 911-212 workmandays, the answer. 









Dynamical Formulas. 


263 


Dynamical Formulas for Uniform Motion. 

Space in Feet passed through in the Time T. 

Sr=VT. 

s=ZI. 

p 550 T H 



F 

F 

F 

F= P . 

Force or Pressure in Pounds. 
jp _550 11 p_ K 

p 1980000& 

V 

V 

S 

S 

V— s . 

Velocity in Feet per Second, 
ir- P v _550 H 

V— K 

T 

F 

F 

F T 

rp _ S 

Time of Duration in Seconds. 
rp_ s F T _ F S 

T _ K 

V ' 

P 

550 H 

F V 

P — F V. 

Power in Effects. 

P — F S . P — 550 H. 

ii 


T 

Horse-power. 

w- FV . n- F s . 

T 

B- K . 

560 

550 

550 T 

550 T 

K = FVT. 

Work in Footpounds. 

K — P T. K=FS. 

K = 550 H T. 

FVT 

Workmandays. 

FS . pT 

k- HT . 

1980000 

1980000 

1980000 

3600 

y _2 7r r n 

Circular 

60 V 

n — 

’ Motion. 

60 V 

r — 

S — 2 7T r N. 

60 

2 7r r 

2 7r n 


p F 2 7 x r n 

tt F2rrrn jj F r n 

K = F2irrN. 

60 

550X60 5250 

p_ 5250 H 

r _60 N ' 

5250 H 

v — 

B — K . 

r n 

n 

Fr 

F 2 7T r 

n — revolutions per minute. 

r = radius of the circle in feet. 

N — total no. of revolutions. 

7T = 3-14159265. 
































264 


Observed Results of Power. 


OBSERVED RESULTS ( 

)F POWER 

ft 


Work- 


Veloo’y 

V 



Description of Works. 

hrs. per 
day. 

Force. 

F 

Effects. 

P 

Horses. 

H 

A man can raise a weight by a single fixed 



40 

0.072 

pulley,. 

6 

50 

0.8 

A man working a crank, .... 

A man on a tread-wheel (horizontal), 

8 

20 

2.5 

50 

0.090 

8 

144 

0.5 

72 

0.130 

A man in a tread-wheel (axis 24° from ver¬ 
tical), . 

8 

30 

2.3 

69 

0.125 

A man draws or pushes in a horizontal 
direction,. 

8 

30 

2 

60 

0.109 

A man pulls up or down, .... 

8 

12 

3.7 

44.4 

0.080 

A man can hear on his hack, 

7 

95 

2.5 

237.5 

• . . 

A horse in a horse-mill, walking moderately, 

8 

106 

3 

318 

0.577 

A horse in a horse-mill, running fast, 

An ox in a horse-mill, walking moderately, 

5 

72 

9 

648 

0.165 

8 

154 

2 

308 

0.558 

A mule “ “ 

8 

71 

3 

213 

0.30S 

An ass “ “ “ 

8 

33 

2.65 

87.4 

0.160 

Oil toad Foot-roads, like those in 






Peru. 






A man can hear,. 

10 

50 

3.5 

175 


Llama of Peru can hear, .... 

10 

100 

3.5 

350 


Donkey can hear,. 

10 

200 

3.5 

700 


Mule can hear,. 

10 

400 

5 

2000 


Flour Mills, 






For every 100 pounds of fine flour ground per hour, require, 


550 

1.000 

One pair of mill-stones of 4 feet diameter, making 120 revolutions 
per minute, can grind 5 bushels of wheat to fine flour per hour, 

2400 

4.36 

One pair of mill-stones of 4 feet diameter, making 120 revolutions 
per minute, can grind 5 bushels of rye to coarse flour per hour, 

1600 

2.91 

Saw Mills, alternative. 





For every 20 square feet sawed per hour, in dry oak, there re- 



quires, .. 




550 

1.000 

Dry pine, 30 square feet per hour, . 

• 

r • 

• 

550 

1.000 

Circular Saw. 






A saw 2.5 feet in diameter, and making 270 revolutions per 



minute, will saw 40 square feet in oak per hour, with 

• 

• • 

550 

1.000 

In dry spruce, 70 square feet per hour, 

• 

• • 

• 

650 

1.000 

Threshing Machine. 





Velocity of the feed-rollers At the circumference, 

0.55 feet per 



second. Diameter of threshing-cylinder 3.5 feet and 4| feet long, 



making 300 revolutions per minute, can thresh from 30 
bushels of oats, and from 25 to 35 bushels of wheat, per hour 

to 40 

2200 

4.000 

One man with a flail can thresh half a bushel per hour (wheat), 

70 

0.127 

Rolling Mills. 






Bar iron-mills. Two pair of rough rollers, two pair of finishing 
rollers, six puddle furnaces, two welding furnaces, making 10 tons 



of bar iron per 24 hours, rollers making 70 revolutions per minute, 



require,. 




29000 

80 

Plate-mill requires about five horses per square foot of plates 
rolled. Largest size plate rollers should not make over 30 revolu- 



tions per minute. 






















Dredging Machines. 


2 r,r 


DREDGING MACHINES. 


Letters denote. 

T = tons of materials excavated 
and raised per hour. 
h = hight in feet in which the ma¬ 
terials are raised above the 
bottom of the excavated 
channel. 

k — O’l for hard clay with gravel. 
k = 0‘07 for hard pure clay. 
k = 0*i >6 for common clay or sand. 
k = 0*04 for soft clay or loose sand. 
k = 0*03 for very loose materials. 
H = horse power required for ex¬ 
cavating and raising the ma¬ 
terials. 

F = force in pounds required to 
feed the Dredge ahead. 
v = velocity of the buckets in feet 
per second. 


Formulas. 


H—T 


(too+*) 


T = 


700 H 
ft+700 


F = 


650 II 


v 


F = 


560 Tk 
v 


T 700 


Example 1. What power is required to excavate T=160 tons of hard 
pure clay per hour, and raise it up h =25 feet above the bottom of the 
channel! For hard clay fc=o-07. 

25 

H —160 ( -1-0• 07) = 16-9, or 17 horses. 

' 700 ‘ ' ’ 

Example 2. What force F=1 is required to feed the Dredge ahead lor 
the above example when the buckets move v=l foot per second. 

F= 15 ^—'Jt = 9295 pounas. 

LEATHER BELTS. 

Letters denote. 

b = breadth of leather belt in inches, 

H= horse power transmitted by the belt. 
v = velocity of the belt in feet per second. 

n Z revoltittoi” Statute } of the 8malle8t belt 

F = force in pounds transmitted by the belt. 
a = number of degrees occupied by the belt on the small pulley. 

safe working strength in pounds per inch of width of belt, which, for 
oak-tanned leather £ inch thick, cemented and riveted joint, can be taken 
at 100 pounds, and less in proportion for weaker belts. 


30,000,000 AT 
dnaS ’ 

130,000 H 


F- 


v aS 

126,500 II 


d n 


H= 


dnF 


11 = 


126,500 

bdnaS 


11 = 


30,000,000 ’ 

bvaS 


130,000 


6 


Example. A leather belt is to transmit H=lb horse power over a pulley 
d= 36 inches in diameter, making n= 80 revolutions per minute, angle of 
contact a = 170°, and the safe working strength S= 100 pounds per inch of 
width. Required the width of the belt. 


Formula 1. 


Width b = 30,000,000 X 75 

36 X 80 X 170 X 100 


: 46 inches, nearly. 


























263 


Friction. 


FRICTION. 


I The resistance occasioned by Friction is independent of the velocity of mo¬ 
tion ; but the re-effect of friction is proportional to the velocity. Friction is in¬ 
dependent of the extent of surface in contact when the pressure remains the 
same, but proportional to the pressure. This law was established from experi¬ 
ments by Arthur Morin in the years 1831-32 and 1833, from which a summary 
is contained in the accompanying Table. 

Letters denote. 

a = Fibres of the woods are parallel to themselves, and to the direction of 
motion. 

b = Fibres at right-angles to fibres. 

c — Fibres vertical on the fibres which are parallel to the motion. 

d = Fibres parallel to themselves, but at right-angles to the motion, length 
by length. 

e = Fibres vertical, end to end. 

Example. A vessel of 800 tons is to be hauled up an inclined plane, which 
inclines 9° 40' from the horizon; the plane is of oak, and greased with tallow. 
What power is required to haul her up ? 

The coefficient for oak on oak with continued motion is f — 0-097, say 0T, 
then, 

800Xsin.9° 40' = 800X0-16791 = 124-328 tons, 
the force required if there were no friction, and 

800Xcos.9° 40'XO’l = 800X0-9858X0-1 = 78-864 tons, 
the force required for the friction only, and 
134-328 
78-864 

213-192 tons, the force required to haul her up. 

The effect lost by friction in axle and bearings is expressed simply by the 
formula 


P = 


Tt d Wnf 
12-60 = 


Wd nf 
230 > 


in which TP = the weight of pressure in the bearing, d = diameter on which 
the friction acts in inches, n = number of revolutions per minute, and/ = co¬ 
efficient of friction from the Table. In common machinery kept in good order 
the coefficient of friction can be assumed to f = 0"065, then 


r= 


W dn 
~353J : 


Wdn_ 
~ 1941500 


Example. The pressure on a steam-piston is 20000 pounds, and makes n — 40 
double strokes per minute. Required the friction in the shaft of d = 8 inches ? 

_ 20000X8X40 

U — —i941500~ = Worses, the loss by friction. 

Friction in Guides. 

TP= pressure on the steam piston in pounds. 

S = stroke of piston in feet. 

I — length o t connecting rod in feet. 

H= horse power of the friction. 


H 


W S n 


350000 


Example. The pressure on a steam piston being W = 30.000 nounrts strntra 
\ feet, length of connecting rod l = 7 feet, and making^ 5<S SS4S2 
minute. Required the horse power of the friction H = ? ^ 1 

jj 30000X4X50 ,. 10 l 

M = 35006075X7*=4^ h ° rSeS * 


I 






















Friction, 


267 


TABLE OF FRICTION FOR PLANE SURFACES IN CONTACT. 


Kind of Materials in contact. 


Imbricated 

with. 

Ooeffi.cn 

Motion. 

cnt in 
Starting. 

Oak on Oak, ... 


a 

0 

0-478 

0-625 

« «... 


99 

tallow 

0-097 

0-160 

« «... 


99 

lard 

0-067 

.... 

« «... 


6 

0 

0-324 

0-540 

« «... 


99 

nnctuous 

0-143 

0-314 

« «... 


99 

tallow 

0-083 

0-254 

« «... 


99 

water, 

0-25 

* • . . • 

« «... 


d 

0 

0-336 

• • • • 

“ «... 


0 

0 

0-192 

0-271 

« «... 


e 

0 

• • • • 

0-43 

Cast-iron on Oak, 


a 

0 

0-400 

0-570 

66 66 . . 


99 

soap 

0-214 

• • • • 

a • 


99 

tallow 

0-078 

0-108 

Wrought-iron on Oak, 


99 

0 

0-252 

• • • • 



99 

tallow 

0-078 

.... 

Wrought iron, together, - 


a 

0 

0-138 

0-137 

66 66 m 


a 

unctuous 

0-177 

• • . • 

66 66 m 


99 

tallow 

0*082 

.... 

66 66 m 


99 

olive oil 

0-070 

0-115 

Wrought on cast-iron, 


a 

0 

0-194 

0194 

66 66 m 


99 

unctuous 

0-18 

0-118 

66 66 


99 

tallow 

0-103 

0-10 

66 66 m 


99 

olive oil 

0-066 

0100 

Cast-iron on cast-iron, 


a 

water 

0-314 

0-314 

66 66 m 


99 

soap 

0-197 

.... 

66 66 


99 

tallow 

o-ioo 

o-ioo 

66 66 


99 

olive oil 

0-064 


Wrought-iron on brass, - 


a 

0 

0-172 


66 66 m 


99 

unctuous 

0-160 


66 66 


99 

tallow 

0-103 

• • • • 

66 66 


99 

lard 

0-075 


66 66 


99 

olive oil, 

0-078 

• • • • 

Cast-iron on brass, 


a 

0 

0-147 


66 66 _ 


99 

unctuous 

0-132 


66 66 . . 


99 

tallow 

0-103 

• • « • 

66 66 _ 


99 

lard 

0-075 


66 66 m 


99 

olive oil 

0-078 


Brass on brass, - • 


a 

0 

0-201 

• • • • 

« «... 


99 

unctuous 

0-134 

• • • • 

« «... 


99 

olive oil 

0-053 


Steel on cast-iron, - • 


99 

0 

0-202 

• • • • 

66 66 m 


99 

tallow 

0-105 


66 66 m 


99 

lard 

0 081 

• • • • 

66 66 


a 

olive oil 

0-079 

, , 


FRICTION OF AXLES IN MOTION. 




Oil, Tallow, or Hog’s Lard. 


Dry or slightly 

Supplied in the 

The grease 

Designation of surface in 

greasy, or wet. 

ordinary 

continually 

contact. 


manner. 

running. 

Brass on Brass, ... 


0079 


« on cast-iron, 


0-072 

0049 

Iron on Brass, ... 

0-251 

0-075 

0-054 

“ on cast-iron, 

Cast-iron on cast-iron, 


0-075 

0-054 

0-137 

0-075 

0-054 

“ on Brass, - 

0-194 

0-075 

C-054 

Iron on lignum-vitae, 

0-188 

0-125 


Cast-iron on « 

0.185 

o-ioo 

0-092 

Lignum-vjtac! on cast-iron, 

. 

0116 

0170 




































2G8 


Strength of Materials, 


STRENGTH OF MATERIALS. 

Table I., shows the weight a column can bear with safety; when the weight 
presses through the length of the column. The tabular number is the weight 
in pounds or tons per square inch on the transverse section of a column of 
a length less than 12 times its smallest thickness. 


Table I* 

RESISTANCE FOR COMPRESSION. 


174. 


Kind of Materials 

Oak, of good quality, 

Oak, common, 

Spruce, red (Sapin rouge), 

“ white, (Sapin blanc 
Iron, wrought, 

Iron, cast, 

Basalt, 

Granite, hard, 

“ common, 

Marble, hard, - 
“ common, 

Sandstone, hard, 

“ loose, 

Brick, good quality, 

“ common, 

Lime-stone, of hardest kind, 

“ common, 

Plaster-Paris, - 
Mortar, good quality, and 18 months old, 
Do. common, - 


Pounds. 

432 

2S0 

540 

140 

14400 

28750 

2875 

1000 

575 

1435 

431 
1295 

5-6 

175 

58 

720 

432 
86 
58 
36 



When the length or height of the column is more than 12 times its smallest 
thickness, divide the tabular weight by the corresponding number in this 
Table. 


LengthXthickness 

12 

18 

24 

30 

36 

42 

48 

54 

60 

Divide by 

r» 

1-6 

2 

2-8 

4 

5 

6 

8 

12 


Example. A building which is to weigh 2000 tons is to be supported by piles 
of Sapin rouge Spruce 18 feet in length, and 12 inches diameter. How many piles 
are required to support the building ? 


12*X0-785X0’241 

16 


17 tons, the weight which each pile can bear, 


and 


2000 T10 .. 
-yy • = 118 piles. 


Professor llodgkinson’s Formulse for Crushing Strength of 

Cast Iron Pillars. 

The ends of the pillars should be perfectly flat and square, and the load to 
hear even on the whole surface. 

T=crushing weight in tons. 

D=outside and d inside diameters in inches. 

I ^length or height of pillar in feet. 










































Cast Ikon Pillar* 


269 

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HI 

c* 

£ ° 
u- 

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^ ii 2 M ^ ° ^ ^ ° cocoiococs 

^ ‘ COCOCSIC^CS MHHHH 

§ : rcj i2 S 52 00 ^ ^ 10 ^ r-i o co o <cq o o o c o 

J ?! 2 2 ? OOil—‘u^r-H CO lO CO (N C O CO CO H O 

HCQ HHCC lO CO CO (M CS1 rH r-H rH ?—i p—| Qi CO N N CO 

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0 pva pvT 22 2? ^ CO ^ OO Hi hH iO<MN005 

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These values are about half of that by Prof. Hodgkinson’s formula. The points after the numbers mean ciphers. 







































































?70 


Strength of Materials. 


Table II. 

COHESIVE STRENGTH PER SQ. INCH OF CROSS-SECTION. 



1 Just tear asunder. 

With safety. 

Kind of Material*. 


Pounds. 

Tons. 

Pounds. 

Tuns 

Cast Steel, 

. 

134256 

59-93 

33600 

14-98 

Blistered Steel, 

• 

133152 

59-43 

33300 

14-86 

Steel, Shear, - 

. 

128C32 

56-97 

32160 

14-24 

Iron, Swedish bar, - 

• 

65000 

29-2 

16260 

7*3 

“ Russian, 

• 

59470 

26-7 

14900 

6-7 

“ English, 

- 

56000 

25-0 

14000 

6-2o 

“ common, over 2 in. sq., 

36000 

16-00 

9000 

4-0 

“ sheet, parallel rolling, 

40000 

17-85 

10000 

4-46 

“ at right angles to roll, 

34400 

15-35 

8600 

3-84 

Cast iron, good quality, 

- 

45000 

20-05 

11250 

5-00 

“ inferior, - 

- 

18000 

8-03 

4500 

2-0 

Copper, cast, - 

• 

32500 

14-37 

8130 

3-6 

“ rolled, 

- 

61200 

27-2 

15300 

6-8 

Tin, cast, - 

. 

5000 

2-23 

12500 

0-56 

Lead, cast, 

• 

880 

0-356 

220 

0-09 

“ rolled, - 

• 

3320 

1-48 

830 

0-37 

Platinum, wire, 

- 

53000 

23-6 

13250 

5-9 

Brass, common, >- 

* 

45000 

20-05 

11250 

5-0 

Wood. 






Ash, ... 


16000 

7-14 

4000 

1-87 

Beach, ... 


11500 

5-13 

2875 

1-28 

Box,... - 


20000 

8-93 

5000 

2-23 

Cedar, ... 


11400 

5-09 

2850 

1-27 

Mahogany, 


21000 

9-38 

5250 

2-34 

“ Spanish, 


12000 

5-36 

3000 

1-34 

Oak, American white, 


11500 

5-13 

2875 

1-28 

“ English “ 


10000 

4-46 

2500 

1-11 

“ seasoned, - 


13600 

6-07 

3400 

1-52 

Pine, pitch, 


12000 

5-35 

3000 

1-34 

“ Norway,- 


13000 

5-8 

3250 

1-45 

Walnut, - 


7800 

3-48 

1950 

0-87 

Whalebone, 


7600 

3-40 

1900 

0-85 

Hemp ropes, good, - 


6400 

2-86 

2130 

0-95 

Manilla ropes, 


3200 

1*43 

1100 

0-49 

Wire ropes, 


38000 

17 

12600 

5-36 

Iron chain, 


65000 

29 

21600 

9-38 

“ with cross pieces, 


90000 

40 

30000 

13-4 


175. 



To Find the Cohesive Strength. 

Rule. —Multiply the cross-section of the materials in square inches hy the 
tabular number in Table II., and the product is the cohesive strength. 

Example. An iron-bar has a cross-section of 2-27 sq. in. How many tens are 
required to tear it asunder, and how many pounds can it bear with safety ? 

English iron 2-27X25 = 56-75 tons, which will tear it asunder, sad it will bear 
with safety 

2-27X14000 = 31780 pounds. 





























Chains, Hemp and Wire Ropes. 271 


Safety 

Iuches and lGtlis. 

Wht. 

per fathom. 

Price per fathom...mtimate 1 

proof. 

Chain. 

Hemp. 

Wire. 

Chain. 

Hemp. 

Wire. 

Chain. 

Hemp, 

Wire. f 

Strain. 

C«'t. 

Diam. 

Circ’m. 

Circ'rn. 

Pounds 

Pounds Pounds 

$ cts. 

$ cts. 

$ cts. 

Cwt 

1.3 

1 

0-10 

0-4 

0-23 

0-08 

0-06 

0-15 

0-06 

0-08 

2 6 

4-5 

2 

1.6 

0-8 

0-93 

0-47 

0-24 

0-25 

0-12 

0-15 

9 

10 

3 

2-1 

0-12 

2-11 

1-06 

0-54 

0-36 

0-17 

0-22 

20 

18 

4 

2*12 

1-1 

3-75 

1-89 

1-10 

0-48 

0-25 

0-32 

35 

28 

5 

3*7 

1-6 

5-86 

2-94 

1-83 

0-60 

0-33 

0-43 

55 

40 

6 

4-2 

1-10 

8-45 

4-52 

2-56 

0-96 

0-42 

0-54 

80 

55 

7 

4-15 

1-14 

11-5 

6.09 

3-42 

1-25 

0-48 

0-62 

109 

69 

8 

5-8 

2-2 

15-0 

7-55 

4-39 

1-44 

0-60 

0-78 

138 

80 

9 

6-3 

2-6 

18-8 

9-56 

5-48 

1-80 

0-76 

0-90 

160 

94 

10 

6*14 

2-11 

23-0 

11-8 

7-00 

1-86 

0-95 

1-20 

218 

109 

11 

7-9 

2-15 

27-7 

14-3 

8-38 

2-16 

1-14 

1-50 

187 • 

127 

12 

8-4 

3-3 

33-0 

17-1 

9.90 

2-43 

1-37 

1-80 

254 

147 

13 

8-15 

3-8 

38-5 

19-9 

11-9 

2-70 

1-60 

2-10 

293 

168 

14 

9-10 

3-12 

44-7 

23-1 

13-6 

3-06 

1-85 

2-28 

335 

199 

15 

10-5 

4-1 

51-1 

26-3 

16-0 

3-70 

2-10 

2-45 

397 

220 

1 in 

11- 

4-6 

58-0 

30-2 

18-6 

4-33 

2-42 

2-73 

440 

246 

1-1 

1111 

4-11 

65-6 

34-1 

21-3 

4-68 

2-73 

3-10 

492 

278 

1-2 

12-6 

5 in. 

73-7 

38-2 

24-2 

5-58 

3-06 

3-50 

545 

302 

1-3 

13-1 

5-5 

82-1 

42-6 

27-4 

5-86 

3-40 

3-91 

604 

332 

1-4 

13-12 

5-10 

91-0 

47-1 

30-7 

6-42 

3-77 

4-35 

663 

365 

1-5 

14-7 

6 ln 

100 

52-0 

35- 

7.08 

4-16 

4-89 

730 

399 

1-6 

15-2 

6-5 

110 

57-1 

38-7 

7-75 

4-57 

5-35 

798 

435 

1-7 

15-15 

6-10 

120 

634 

42-6 

8-42 

5-07 

5-86 

869 

472 

1-8 

16-8 

6-15 

131 

67-9 

46-7 

9-15 

5:44 

6-35 

944 

553 

1-10 

17-14 

7-10 

154 

79-8 

56-4 

10-07 

6-38 

7-63 

1105 

638 

1-12 

19-4 

8-4 

178 

92-6 

66-0 

12-38 

7-40 

8-83 

1275 

729 

1-14 

20-10 

8-14 

205 

106 

76-5 

14-15 

8-48 

10-00 

1457 

825 

2 m 

22- 

9.8 

232 

121 

88-0 

16-00 

9-70 

11-50 

1650 

1072 

2-4 

24-12 

10-12 

293 

153 

112 

20-75 

10-25 

14-60 

2141 

1288 

2-8 

27-8 

12 in. 

363 

189 

140 

25- 

15-10 

18-00 

2575 

1559 

2-12 

30-4 

13-4 

438 

229 

172 

30-25 

18-30 

21-80 

3117 

1854 

3 in 

33- 

14-8 

522 

272 

205 

36-00 

21-80 

25-90 

[3708 


The prices of the chains are taken from that in England and added 50 
per cent. Price of hemp ropes from Weaver, Fitler & Co., Rope manu¬ 
facturers, Philadelphia. The prices of Wire ropes are deduced from 
the price list of John A. Roebling, Patent Wire Rope Manufacturer, 
Trenton, N. J. 

The Safety proof is here taken one half of the ultimate strength which 
may be trusted on for new ropes, but when much r>n]r one nuarter 

or less should be trusted upon for safety. 
































272 


Strength of Materials. 


LATERAL STRENGTH OE MATERIALS. 

The formulas for lateral strength are here reduced to the simplest pos¬ 
sible form, and are in consequence subject to conditions which must be 
particularly attended to. In calculating the strength of beams of ir¬ 
regular sections as shown by the figures 210 to 217 on page 173, it is neces¬ 
sary to maintain the proportions marked on the figures and the calcu¬ 
lation will be correct. For the sections 206 to 209 any proportion will 
answer in the formulas. The weight of the beam itself has not here 
been taken into consideration, for which allowance must be made if 
considerable. 

Letters denote. 

I = length of beam in feet. See figures. 

h = height, 6=breadth or thickness in inches of the beam, where the 
strain is acting. 

k = coefficient for the different materials and sections of beams, to be 
found in the tables. 

x — modulus of ela-sticity of materials. See Table. 

/= elastic deflection in inches. 

W— weight in pounds which the beam can bear with safety, being 
about one quarter of the ultimate strain at which the beam 
would break. 

Example 1. Fig. 200. A rectangular beam of oak fastened in a wall 
projects out 1=6 feet 4 inches, h =8 inches, and 6=5 inches. Required 
what weight it can bear on the end W =>? 


W= 


30X5X8 2 
6 333 ~ 


= 1509 pounds, with perfect safety. 


Example 2. Fig. 201. A beam of section fig. 211, with thickness 6=1'25 
inches, height 6=22-5 inches, supported at the two ends in a length 1=26 
feet. Required what weight W=1 it can bear in the middle. For cast 
iron coefficient 6=260. 


W 


4X260X1 -25X22-5 3 


25 


= 26325 lbs.=11-8 tons nearly. 


Example 3. Required the elastic reflection for the same beam and con¬ 
dition as in the foregoing example 1 See Table, modulus of elasticity 
#=2285 for cast iron. See page 276. 


/ = 


26325X25 3 


16X2285X1-25X22-5 3 


0-80 inches, nearly. 


Example 4. Fig. 204. A wrought iron girder of section fig. 217, consist¬ 
ing of four angle irons of a= 3-5X0'5X 2 X 4 =14 square inches, the plate 
being 0-5:1-35=0-37 inches thick, and 6=18 inches deep by 1=22 feet. Re¬ 
quired how much weight evenly distributed the girder can bear with 
safety 1 

W= 8 X800X14X 18 —73309 lbs.=32*75 tons. 

22 

If plates being riveted to the angle iron at top and bottom, add that 
area to a. 

Example 5. Fig. 222. The crank R= 3-5 feet, force F=3860 lbs., length 
of the shaft 1=64 feet, diameter Z)=5-25 inches. Required the twisting 
in degrees. The shaft being of wrought iron for which #=4110. 


Degrees 


425X3860X3-5X64 


4110X5-25 4 


= 11-76°. 













Strength and Elasticity of Materials. 


273 

















































































































274 


Different Forms of Beams.. 


■ 

206. 

Coefficient £. 

Cast iron, 150 
Wro’tiron, 120 
Wood, 30 1 

a H A 

.Ir 1 

212. 

Coefficient k. 

Cast iron, 236 
Wro’tiron, 189 

♦ 

207. 

Coefficient £. 
Cast iron, 150 
Wro’t iron, 120 
Wood, 30 

6 &=&. 

T> 

A | 4 

sl Li 

1 

< 6h > 

213. 

Coefficient Jc. 

Cast iron, 250 
Wro’tiron, 200 


208. 

Coefficient £. 
Cast iron, 88 
Wro’t iron, 70 
Wood, 18 

b h~=D 3 . 

<5 lf>, T 

A 

lolwzl 

214. 

Coefficient k. 

Cast iron, 700; 
Wro’t iron, 560 


1 

e 

209. 

Coefficient £. 
Cast iron, 88 
Wro’t iron, 70 
Wood, 18 

b A 5 =Z> 3 — cf. 


215. 

Coefficient k. 

Cast iron, 900 


210. 

Coefficient £. 

Wro’tiron, 700 

b Ji > =a h. 

' IO ' 

216. 

Cast iron tube, 

£=800. 

^TjPJ 
_j 3 js.L. 

| ^^<6 l »• 

211. 

Coefficient Jc. 

Cast iron, 260 
Wro’tiron, 208 

A Jo 

217. 

£=800, 
b A 3 =a Ji, 
a=area of all 
the four angle 
irons in square 
inches. 













































Strength of Materials. 


275 



218. 

A beam fixed in one end and loaded 
at the other, should have the form of a 
Parabola, in which 1= abscissa and h — 
ordinate. y= depth, x= length from W. 




219. 


W= 


kbh* kb l i s 


l cos.v 


b> 


w 


* /* // l \\5 K 


220 . 


W= 


36 k b h* 


l 


Divide the length into 24 equal parts, 
place 14 in the middle and 5 at each end. 



221. To cut oat the stoutest rectangular 
beam from a log. 

1st case, divide the diameter in 3 equal 
parts, and draw lines at right-angles as 
represented. 

2d, divide the diameter in 4 equal parts. 

1, 6= 1*414 b, non-elastic. 

2, h—l ,Y J3 b , elastic beams. 


. 3 /FR 

=w—= 


80 


II 


. , 425 FR l 

Twisting m degrees =-——. 

x I) 



223. _ 

D : d—-tyR : ig/r, 


D= 80 


H 


Twisting in degrees = 


2233000 H 


i) 


xnD 4 









































































276 


Strength of Materials. 


Absolute ami Ultimate Strength, of Materials. 


Kind of Materials. 

Wrought iron, . 



Safety. 

120 

Coefficie 

Inter. 

162 

nt k. 

Pr. cir. 

240 

Ultimate. 

488 

Elasticity. 

X 

4110 

Cast iron, 



150 

200 

300 

600 

22S5 

Cast steel, soft,. 

• 

• 

385 

619 

170 

1540 

4300 

Cast steel, hardened, 


• 

1050 

1400 

2100 

4200 

6000 

Blistered steel, soft, . 

« 

• 

175 

233 

350 

700 

4200 

13 mss, • 0 • • 



58 

75 

113 

226 

1280 

Copper, 

. 


53 

71 

106 

212 

2160 

Zinc, • • • • 



15 

20 

30 

61 

2360 

Tin, .... 
LcBid, .... 

• 

. 

17 

23 

34 

69 

... 


• 

4 

6 

9 

18 

100 

Ash, .... 

• 

. 

45 

56 

85 

170 

221 

Hickory, 



67 

90 

135 

270 

... 

Chestnut, sweet, 

. 


42 

56 

85 

170 

... 

Oak, white, . , 



50 

66 

100 

200 

300 

Oak, English, . 

• 


25 

33 

50 

100 

248 

Canadian oak, 



37 

49 

73 

147 

283 

Pine, white, 

• 


34 

45 

67 

135 

... 

Yellow pine, . 



38 

50 

75 

150 

268 

Teak, 

. 


51 

68 

102 

205 

316 


The absolute safety weight is here taken one-quarter of the ultimate breaking 
weight, but when the weight is acting at short intervals one-third might be relied 
upon, or in pressing circumstances one-half, when the materials in the beams are 
known to be of good quality; but the latter never to be exceeded. 

Properties of some South American Woods, 

Taken from, the borders of the rivers Perene and Madre de Dios, and experi¬ 
mented upon by the author of this Pocket-Book. 


Peruvian Names of the 
Woods. 

Chonta (Palm), . . . 

Balshmo,. 

Shacaranda, .... 
Jebe (Ind.-rubber tree)* 

Amarillo,. 

Oaoba,. 

Huachapeli, .... 

Nogal,. 

Jebe(best Ind.-rubber)* 
M. Barigon, .... 



Specific 

Wt. per 

Hard- 

Ultimate 

Elas- 

Color. 

gravity. 

cub. foot, 
lbs. 

ness. 

H 

strength. 

k 

ticity. 

X 

Black, . . . 

1.564 

96.75 

28 

450 

640 

Brown, . . 

1.207 

75.25 

22 

422 

492 

Brown stripes, 

0.991 

61.75 

18 

343 

322 

Light yellow, 

0.797 

49.65 

15 

351 

305 

Yellow, . . 

0.734 

45.75 

13 

334 

300 

Light brown, 

0.613 

38.20 

11 

128 


Oak,. . . 

0.566 

35.25 

10 

134 

180 

Dark brown, 

0.551 

34.35 

10 

131 

158 

White, . . . 

0.527 

32.85 

9 

162 

262 

White, . . 

0.282 

17.58 

6 

62 

92 


* There are different kinds of trees which give India-rubber, but of different 
quantity and quality. 

The woods were perfectly dry. Four experiments on each were made. 

The hardness, H, is compared with that of substances on page 333. 

The coefficient, k, is the ultimate lateral strength of the woods. 
x = modulus of elasticity determined near the ultimate strength. 


k = 


Wl 


and % = 


Wl 3 


4 b h? 16 fb h 3 

Meauing of letters is the same as that on page 272. 


Fig. 201, p. 273. 

































Pile Driving. 


277 


PILE DRIVING-. 

Letters denote. 

M ■= weight of the ram in pounds. 

S — fall of the ram in inches. 
in — weight of the pile in pounds. 
s — space in inches which the pile sinks by the blow. 
r ==> resistance of the ground iu pounds. 

a = section area in sq. in. of the pile, sharpened to a point not more 
than 45°. 

k = coefficient for the hardness of the ground. 
h = depth to which the pile is driven. 

W = weight in pounds which a dx*iven pile can bear with safety after 
the last blow when the pile sunk s inches. 

V = velocity in feet per second by which the ram strikes the pile. 

Ram and pilehead considered non-elastic and perfectly hard. 


F = 8yS - 


M 3 S 

r (M-\-mj* 


W= 


M 3 S 

6s(M+my» 


1 

2 


3 


8 M/S 

v = ---— * 

M-\-m 

_ M 3 S 
s (M-j-m) 2 

r—akyh 


4 


5 

6 


Example 1. A wooden pile 18 feet long by 12 inches square, driven 
h —12 feet into common natural ground imbedded with tenacious clay for 
which may be assumed the coefficient A:=50. Required how much the 
pile will set s—1 into the ground at a blow with a ram of 111=3600 lbs. 
falling S=42 inches. 

The weight of the wooden pile will be about m=18X40=720 lbs. 

Area of the pile a=144 square inches. 

Resistance r= 144X50/12 =23840 lbs. 

The resistance sought from this formula 6, cannot be depended upon 
lor calculating the weight the pile can bear with safety. 


The set s 


3500 3 X42 


23840 (3500+720) 3 


=4*23 inches. 


Suppose the set to be s=4-23 inches at the last blow, required what 
weight the pile can bear with safety 1 


\V 


3500 3 X42 

0X4-23 (35004-720)' 3 


3984 lbs. 


This can be depended on with safety, if calculated from the actual set 
of the pile at the last blow. 

For ordinary pile driving a heavy ram and short fall is the most effec¬ 
tive, but in some cases when the ground itself is elastic, or when driving 
piles in pure sand it is found more advantageous to use a high fall of 
the ram. 

Approximate Coefficients. ^ 

In coral formations,.120 

In hard clay with gravel, .... 100 

In hard pure clay,.70 

In common clay or sand, .... 50 

In soft clay or loose sand, . . . .40 

In very loose materials, .... 30 




















278 


Wrought Iron Beams. 


LATERAL STRENGTH. 


For wrought iron beams , letters denote. 


W= weight in tons with safety, uniformly distributed on a beam reel¬ 
ing on two supports. 


S = compressive strain in tons per square inch of top 0-5 


o= section area in square inches of top and bottom flanges of the 
beam. Top and bottom flanges to be alike. 
a'— section area in square inches of web or stem. 
h = height of beam in inches. 

I = length in feet between supports. 

f= deflection in inches of the beam in the centre, when the weight is 
uniformly distributed. 

p = weight in pound per square foot of flooring to be supported by the 
beams, which in ordinary cases is estimated to P=140 lbs. 

B = distance in feet between the beams. 
w = weight of the whole beam in pounds. 


W Z 3 



6 


1 





e 


2 


2240IF 


3 Wl 


B = —— 

PI 


7 


S 


3 



22 w W 


w = 3'384 1 (a-\-a r .) - 4 e - Wl~‘ “ " " 8 

Formula 6 gives the safety deflection of a wrought iron beam, which 
should never be exceeded. 

Example. A flooring of 1 =32 feet by 60 feet long to be constructed to 
support P—175 pounds per square foot. Required what kind of beams 
and how many are necessary ! and what will be the cost of them? 

In the table will be found the nearest star to 32 feet span is a 12 inch 
beam bearing W= 8-71 tons, when the distance between the beams in 
the flooring will be, 


2240X8-71 
175X 32~ 


Formula 7. B — 


= 3-5 feet. 


60 


Number of beams =-- —1—16 about. 

3-5 

Add one foot to each beam for the supports at the ends, and the cost 
will be, 33X16X1 •90=1003-70 dollars. 


The following Table contains sections of iron roiled by the Phoenix 
Iron Company. Office 410 Walnut Street, Philadelphia. 


Rules. 


The price per foot multiplied by 5280 gives the price per mile. 

The weight in pounds per foot multiplied by 2-36 gives the weight in 
tons per mile. 

The price per foot multiplied by 2240 and divided by the weight in 
pounds per foot gives the price per ton. 












Strength of Different Sections of Wrought Iron Beams. 


279 


Strength of different Sections of Wrought Iron Beams 

Made by the Phoenix Iron Company,/'or Sustaining with Safety 
a Load Uniformly Distributed. 


1 

Compound Girders. 

Dis. 

800 

667 

553 

bet. I 


w= — 

* l 

W= T 

sup 

h =18 i. 

h =15 i. 

/i=12 i. 

feet. | 

tons. 

tons. 

tons. 

10 1 

80-00 

66-67 

55-33 

12 | 

66-66 

55*56 

44-44 

14 

57-14 

47-61 

38-09 

16 

50-00 

41-67 

33-33 

18 

44-44 

37-04 

28-52 

20 

40-00 

33-33 

26-66 

22 

36-36 

30-30 

24-24 

2 4 

33-33 

27-77 

22-22 

26 

30-77 

25-64 

20-05 

28 

28-57 

23-80 

19-05 

30 

26-66 

22-22 

17-77 

32 

25-00 

20-83 

16-66 

34 

23-53 

19-60 

15-65 

36 

22-22 

18-52 

14-26 

38 

21-05 

17-37 

14-00 

40 

20-00 

16-66 

13-33 

42 

19-05 

15-87 

12-70* 

44 

18-18 

15-15 

12-12 

46 

17-37 

14-48* 

11-44 

48 

16-66 

13-88 

11-11 

50 

j 16-00* 

13-3.3 

10-66 

Per 

78 lbs. 

1 71 lbs. 

59 6 lbs 

Foot ' 

$4-68 

$4-26 

$3-67 


Solid Rolled Beams. 


490 

W =T 

h= 15 i. 


TP= 
h 


tons. 

49-00 

40-83 

35-00 

30-63 

27-22 

24-50 

22-27 

20-42 

18-85 

17-50 

16-33 

15-31 

14-41 

13-61 

12-90* 

12-25 

11-67 

11-13 

10-66 

10-21 

9-80 


296 

T 

12 i. 


tons. 

29-60 

24-66 

21-14 

18-50 

16-44 

14-80 

13-45 

12-33 

11-38 

10-57 

9-86 

9-25 

8-71* 

8-22 

7-80 

7-40 

7-05 

6-72 

6-43 

6-16 

5-92 


308 

W= T 

h =9 i. 


h 


168 

T 

9 i. 


tons, 

30-80 

25-69 

22-05 

19-25 

17-11 

15-40 

14-00 

12-85 

11-87 

11-00 

10-26 

9-62 

9-06* 

8-55 

8-11 

7-70 

7-34 

7-00 

6-70 

6-42 

6-16 


tons, 

16-80 

14-00 

12-00 

10-50 

9-33 

8-4 0 

7-63 

7-00 

6-46 

6 - 00 * 

5-60 

5-25 

4-94 

4-66 

4-42 

4-20 

4-00 

3-81 

3-65 

3-50 

3-36 


84 


w= 

l 

h = 7 i. 


tons. 

8-40 

7-00 

6-00 

5-25 

4-66 

4-20 

3-81* 

3-50 

3-23 

3-00 

2-80 

2-62 

2-47 

2-33 

2-21 

2-10 

2-00 

1-91 
1-83 
1-75 
1 68 


_ 48 

W=* j- 

h= 6 i. 


tons. 

4-80 

4-00 

3-43* 

300 

2-66 

2-40 

2-18 

2-00 

1-84 

1-71 

1-60 

1-50 

1-40 

1-33 

1-26 

1-20 

1-14 

1-09 

1-04 

1-00 

•96 


47-8 lbs. 40 lbs. 
i $2-40 $1-90 


60 lbs. 29 3 lbs. 20 lbs. 13 3 lbs. 
$2-40 $1-60 $1-00 $0.66 


The above Table gives the weight in tons, sustained by the several 
kinds of beams, uniformly distributed over them as in a floor. The 
weights given are what may be used in practice, being only 9 tons per 
square inch of that part of the metal subjected to a crushing force. 

Under these weights the beams are within the limits of perfect elas¬ 
ticity, and the deflections are therefore in direct proportion to the load. 

If it be intended to apply the entire weight at the centre, the figures 
in the Table must be divided by two ; if at any other point, the weight 
at the centre is to the weight at any other point, as the square of half 
the beam is to the rectangle of the two parts from where the weight is 
applied. The prices are subject to changes of the market and agree¬ 
ment. 

* When the span of the flooring is given, the star in the Table gives 
an approximation to what beam ought to be employed,- for instance, 
1= 38 feet span should have beams of h =15 inches high, able to bear 
^“12-9 tons uniformly distributed. 





































280 Dimensions, Weight, and Price or Rolled Iron. 



























































Spheres. 


281 


Spheres, Balls— 

-Surfaces, 

Capacity and Weight of. 

Diameter. 

Surface. 

Capacity. 

Cast iron. 

Lead. 

Water. 

Inches. 

Sq. inches. 

Cub. Inches. 

Pounds. 

Pounds. 

Pounds. 

1 in. 

3.1416 

0.5236 

0.1365 

0.2147 

0.0188 ■ 

1.125 

3.9760 

0.7455 

0.1943 

0.3062 

0.0264 

1.25 

4.9087 

1.0226 

0.2673 • 

0.4200 

0.0368 

1.375 

5.9395 

1.3611 

0.3550 

0.5579 

0.0490 

1.5 

7.0686 

1.7671 

0.4607 

0.7248 

0.0636 

1.625 

8.2957 

2.2467 

0.5861 

0.9227 

0.0809 

1.75 

9.6211 

2.8061 

0.7326 

1.1528 

0.1050 

1.875 

11.044 

3.4514 

0.8000 

1.4156 

0.1242 

2 in. 

12.566 

4.1888 

1.0920 

1.7180 

0.1508 

2.125 

14.186 

5.0243 

1.3124 

2.0631 

0.1809 

2.25 

15.904 

5.9640 

1.5592 

2.4482 

0.2147 

2.375 

17.720 

7.0143 

1.8334 

2.8811 

0.2525 

2.5 

19.635 . 

8.1812 

2.1328 

3.3554 

0.2945 

2.625 

21.647 

9.4708 

2.4725 

3.8892 

0.3410 

2.75 

23.758 

10.889 

2.8400 

4.4623 

0.3920 

2.875 

25.967 

12.442 

3.2512 

5.1056 

0.4479 

3 in. 

28.274 

14.137 

3.6855 

5.7982 

0.5089 

3.125 

30.680 

15.979 

4.1721 

6.5568 

0.5752 

3.25 

33.183 

17.974 

4.6835 

7.3623 

0.6471 

3.375 

35.785 

20.129 

5.2612 

8.2521 

0.7246 

3.5 

38.484 

22.449 

5.8525 

9.2073 

0.8081 

3.625 

41.282 

24.941 

6.5089 

10.231 

0.8979 

3.75 

44.179 

27.612 

7.2135 

11.323 

0.9941 

3.875 

47.173 

30.466 

7.9556 

12.500 

1.0968 

4 in. 

50.265 

33.510 

8.7361 

13.744 

1.2064 

4.25 

56.745 

40.194 

10.510 

16.482 

1.4470 

4.5 

63.617 

47.713 

12.439 

19.569 

1.7177 

4.75 

70.882 

56.115 

14.666 

23.035 

2.0202 

5 in. 

78.540 

65.450 

17.063 

26.843 

2.3562 

5.25 

86.590 

75.766 

19.810 

31.089 

2.7276 

5.5 

95.033 

87.114 

22.720 

35.729 

3.1361 

5.75 

103.87 

99.541 

26.000 

40.856 

3.5835 

6 in. 

113.10 

113.10 

29.484 

46.385 

4.0716 

6.5 

132.73 

143.79 

37.453 

58.976 

5.1765 

7. 

153.94 

179.59 

46.820 

73.659 

6.4653 

7.5 

176.71 

220.89 

57.587 

90.598 

7.9520 

8 in. 

201.06 

268.08 

69.889 

109.95 

9.6509 

8.5 

226.98 

321.55 

83.839 

131.38 

11.576 

9 in. 

254.47 

381.70 

99.510 

156.55 

13.741 

9.5 

283.53 

448.92 

117.03 

184.12 

16.161 

10 

314.16 

523.60 

136.50 

214.75 

18.850 

11 

380.13 

696.91 

181.76 

285.83 

26.289 

12 

452.39 

904.78 

235.87 

371.09 

32.572 

13 

530.92 

1150.3 

299.62 

471.80 

41.411 

14 

615.72 

1436.7 

374.56 

589.27 

51.721 

15 

706.84 

1767.1 

460.69 

724.78 

63.616 

16 

804.24 

2144.6 

559.11 

879.61 

77.206 

17 

853.96 

2572.4 

670.71 

1055.0 

92.607 

18 

1017.8 

3053.6 

796.08 

1252.4 

109.93 

19 

1134.1 

3591.3 

936.27 

1472.9 

129.29 

20 

1256.6 

4188.8 

1092.0 

1718.0 

150.80 










2S2 


Weight op Rolled Iron, per Foot. 



— -- 




— - - - 








r i7 

^i, 


Bide in 

Weight in 

Side in 

Weight in 

Diameter 

Weight in 

Diameter 

Weight in 

inches 

pounds. 

inches. 

pounds 

in inches. 

pounds. 

in inches. 

pounds. 

A 

0-013 

3S • 

44-418 

T5 

o-oio 

31 

34-886 

4 

0*53 

31 

47-534 

4 

0-041 

34 

37-332 

A 

0-118 

34 

50-756 

S 

T5 

0-119 

34 

39-864 

i 

0-211 

4 

54-084 

i 

0-165 

4 

42-464 

i 

0-475 

44 

57-517 

i 

0-373 

44 

45-174 

4 

0-845 

4± 

61*055 

4 

0-663 

44 

47-952 

t 

1-320 

4S 

64-700 

4 

1-043 

41 

50-815 

1-901 

44 

68-448 

4 

1-493 

44 

53-760 

| 

2-588 

4f 

•72-305 

4 

2-032 

44 

56-788 

i 

3-380 

4S 

76-264 

1 

2-654 

44 

59-900 

14 

4-278 

44 

80-333 

14 

3-360 

44 

63-094 

li 

5-280 

5 

84-480 

H 

4*172 

5 

66-752 

IS 

6-390 

54 

88-784 

H 

5-019 

54 

69-731 

14 

7-604 

5i 

93-168 

li 

5-972 

54 

73-172 

is 

8-926 

5f 

97-657 

H 

7-010 

51 

76-700 

14 

10-325 

54 

102-24 

if 

8-128 

54 

80-304 

is 

11-883 

5| 

106-95 

14 

9-333 

54 

84-001 

2 

13-520 

54 

111-75 

2 

10-616 

54 

87-776 

24 

15-263 

51 

116-67 

2i 

11-988 

54 

91-634 

2i 

17-112 

6 

121-66 

24 

13-440 

6 

95-552 

2f 

19-066 

6i 

132-04 

21 

14-975 

6i 

103-70 

2i 

21-120 

64 

142-82 

2i 

16-688 

64 

112-16 

2S 

23-292 

6| 

154-01 

2f 

18-293 

64 

120-96 

2f 

25-56 

7 

165-63 

21 

20-076 

7 

130-05 

24 

27-939 

74 

190-14 

24 

21-944 

74 

149-33 

3 

30-416 

8 

216-34 

3 

23-888 

8 

169-85 

34 

33-010 

8i 

244-22 

3i 

25-926 

84 

191-81 

3i 

35-704 

9 

273-79 

34 

28-040 

9 

215-04 

3| 

38-503 

10 

337-92 

31 

30-240 

10 

266-29 

3i 

41-408 

12 

486-66 

34 

32-512 

12 

382-21 


Rule for Finding the “Weight of Pipes. 

The diameter of the pipe in inches, measured from inside to outside, mul¬ 
tiplied by the coefficient for the metal, will be the weight in pounds per linear 
foot. 


Lead, . 
Copper, 
Brass, cast, . 
Cast steel, 
Clay, burnt, 


Coefficients. 


0.1005 

0.0989 

0.0S82 

0.0891 

0.0214 


Brass, rolled, 
Iron, rolled, 
Cast iron, 
Tin, rolled, 
Zinc, rolled, 


. 0.0985 
0.0876 
. 0.0811 
0.0821 
. 0.0808 
































WEIGHT PER FOOT, IN POUNDS, OF CAST-IRON CYLINDERS AND PIPES. 


28:1 


Diam. 

0 

Va 


% 

Vi 

% 

% 

Va 

Diam 

0 

00000 

•03804 

15418 

34675 

61669 

96352 

1-3876 

1-8975 

0 

1 

2-5132 

3-1227 

3-9047 

4-6620 

5-5512 

6-5476 

7-5414 

8-7012 

1 

2 

9-8989 

11-145 

12-491 

13-947 

15-419 

16-999 

18-658 

20-392 

2 

3 

22-205 

24*093 

26-059 

28-104 

30-225 

32-420 

34-695 

37-038 

3 

4 

39-544 

41-984 

44-566 

47-227 

49-963 

52-778 

55-629 

58-637 

4 

5 

61-584 

64-807 

68-005 

71-282 

74-537 

78.068 

81-577 

84-848 

5 

6 

88-825 

92-564 

96-380 

100-27 

104-24 

108-29 

112-42 

116-62 

6 

7 

120-90 

125-26 

129-69 

134-20 

138-79 

143-45 

148-19 

153-02 

7 

8 

157*91 

162-88 

168-15 

173-06 

178-29 

183-55 

188-91 

194-34 

8 

9 

199-86 

205-44 

211-11 

216-86 

222-68 

228-57 

234-56 

240-50 

9 

10 

246-73 

252-94 

259-23 

265-59 

272-03 

278-54 

285-13 

291-81 

10 

11 

298-55 

305-38 

312-28 

319-24 

326-28 

333-40 

340-64 

347-92 

11 

12 

355-29 

362-72 

370-23 

377-83 

390-50 

393-26 

401-08 

408-69 

12 

13 

416-98 

425-02 

433-15 

441-39 

449-64 

458-04 

466-46 

475-00 

13 

14 

483*73 

492-24 

501-02 

509-84 

518-77 

527-72 

536-80 

545-94 

14 

15 

528-15 

564-44 

573-81 

583-76 

592-78 

602-36 

612-04 

621-71 

15 

16 

631-64 

641-54 

651-53 

661-58 

671-73 

681-94 

692-24 

702-61 

16 

17 

712-79 

723-59 

734-19 

744-86 

755-80 

766-44 

777-38 

788-35 

17 

18 

799-30 

810-56 

821-79 

838-17 

844-45 

855-86 

867-42 

879-04 

18 

19 

890-70 

902-48 

914-29 

926-23 

938-20 

950-27 

962-42 

974-64 

19 

20 

986-95 

999-30 

1011-6 

1024-3 

1036-9 

1049-5 

1062-3 

1075-0 

20 

21 

1088-1 

1104-2 

1114-6 

1127-3 

1140-5 

1153-8 

1167-2 

1180-7 

21 

22 

1194-2 

1207-8 

1221-5 

1235-2 

1249-1 

1263-0 

1277-0 

1291-1 

22 

23 

1305-2 

1319-4 

1333-7 

1348-1 

1362-6 

1376-9 

1391-7 

1406-4 

23 

24 

1421-5 

1436-0 

1451-0 

1466-1 

1481-0 

1496-1 

1511-4 

1526-7 

24 

25 

1492*1 

1557-5 

1572*1 

1588-7 

1604-4 

1620-2 

1635-8 

1651-9 

25 

26 

1667-9 

1683-9 

1700-1 

1716-4 

1732-7 

1749-1 

1765-5 

1782-1 

26 

27 

1798-7 

1815-5 

1832-2 

1849-0 

1865-9 

1882-9 

1900-0 

1917-2 

27 

28 

1934.4 

1951-7 

1969-1 

1986-5 

2004-1 

2021-7 

2039-4 

2057-2 

28 

29 

2075-1 

2093-0 

2111-0 

2129-1 

2147-2 

2165-4 

2183-8 

2202-2 

29 

30 

2220-6 

2239-2 

2257-8 

2276-5 

2295-2 

2314-1 

2333-1 

2352 0 

30 

31 

2371-1 

2390-3 

2409-6 

2428-9 

2448-3 

2467-9 

2461-3 

2506-9 

31 

32 

2526-6 

2545-7 

2566-2 

2586-2 

2606-1 

2626-3 

2646-4 

2666-7 

32 

33 

2687-0 

2707-4 

2727-8 

2748-4 

2769-0 

2789-7 

2810-4 

2831-3 

33 

34 

2852-3 

2873-3 

2894-4 

2927-3 

2936-8 

2958-1 

2979-5 

3001-0 

34 

35 

3022-5 

3044-2 

3065-9 

3087-7 

3109-5 

3131-5 

3143-7 

3175-5 

35 

36 

3197-5 

3219-4 

3242-0 

3264-3 

3286-9 

3309-5 

3332-2 

3354-3 

36 

37 

3377-8 

3400-4 

3423-3 

3446-6 

3469-5 

3492-7 

3516-0 

3539-2 

37 

38 

3562-9 

3586-1 

3609-6 

3633-5 

3657-0 

3680-9 

3704-8 

3728-6 

38 

39 

3752-2 

3776-8 

3801-0 

3835-1 

3849-7 

3873-8 

3898-3 

3922-8 

39 

40 

3947-7 

3972-5 

3987-0 

4022-1 

4046-9 

4071-6 

4094-1 

4122-3 

40 

41 

4147-5 

4173-0 

4198-4 

4223-8 

4249-2 

4275-0 

4300-7 

4326-5 

41 

42 

4352-3 

4378-4 

4404-1 

4430-2 

4456-6 

4482-6 

4509-0 

4540-5 

42 

43 

4562*2 

4588-6 

4615-2 

4641-9 

4668-6 

4695-6 

4722-7 

4749-7 

43 

44 

4778-7 

4803-7 

4831-1 

4858-4 

4885-7 

4913-3 

4941-1 

4968-7 

44 

45 

4996-3 

5024-0 

5051-9 

5079-9 

5107-8 

5136-1 

5164-1 

5192-4 

45 

46 

5220-9 

5249-2 

5277-8 

5306-3 

5335-0 

5363-6 

5393-5 

5421-4 

46 

47 

5450-2 

5473-1 

5508-4 

5537-6 

5566-8 

5596-1 

5625-6 

5655-1 

47 

48 

5684-6 

5714-1 

5744-1 

5773-9 

5803-7 

5833-5 

5863-7 

5893-8 

48 


A solid cast-iron cylinder 42% in. diameter weighs 4482-6 pounds per foot. 
Subtract inside cylinder 40 % in. diameter weighs 3972-5 “ “ 

Weight of pipe 1 ^ in* thick will he 510-1 “ “ 














284 


Weight of Flat Rolled Iron per Foot. 


cq 


o 

o 

fe 

h 

Q 

ft 

e 

o 

* 


rH 

0 

k5 

3 

fe 

0 

+» 

43 

M) 


JS 


c3 


cS 


fl 

c3 

*• 

CO 

CO 

a> 

ft 

o 

• rH 




o w 

H fX 

o> ^ 


<« Pf P, 

J -8 8. 


a eg 

* § 


CO 

rH 

tO 

05 


CO 

CO 

1H 

<M 

CO 


CO 

cq 

CO 

o 

^T< 

CO 

cq 


CO 

o 

<o 

05 

CO 

CO 

b» 

b- 

CO 

CO 

to 

o 


05 

co 

co 

cq 

CO 

rH 

CO 

to 

• 

o 

05 

b- 

• 

CO 

to 


CO 

CN 

rH 

• 

o 

to 

* 

• 

CO 

• 

cp 

cq 

• 

cq 

• 

rH 

• 

tH 

• 

to 

• 

O 

• 

cq 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

<H 

05 

CO 

b- 

cb 

tO 

hH 

CO 

cq 

rH 

rH 


CO 

uO 

tO 

tO 

'H 

Th 


co 

co 

co 

to 

cq 

05 

CO 

co 

05 

CO 




o 

O 

O 

O 

O 

O 

O 

O 

o 

cq 

cq 

cq 

rH 

rH 

pH 

o 

® 

® 

® 

cq 

00 

• 

b- 

• 

CC 

to 

• 

CO 

<bl 

rH 

• 

o 

o 

o 

• 

o 

• 

o 

• 

05 

• 

o 

• 

o 

• 

® 

• 

tO 

• 

o 

• 


rH 

rH 

rH 

rH 

rH 

rH 

H 

rH 

rH 

05 

CO 

b- 

CO 

to 


CO 

cq 

rH 

rH 



CO 

rH 

CO 

rH 

CO 

O 

tO 

tO 


hH 

CO 

CO 

cq 

cq 

rH 

® 

tO 

CO 



rH 

Cq 

<M 

CO 

CO 



o 

to 

o 

to 

o 

tO 

o 

tO 

® 

cq 

05 


cq 

• 

CO 

tO 

• 

H* 

CO 

oq 

• 

rH 

• 

O 

to 

• 

to 

• 

CO 

• 

co 

• 

• 

• 

CO 

• 

CO 

• 

05 

• 

• 

oo 



rH 

rH 

rr 

rH 

rH 

rH 

rH 

05 

co 

b>- 

CO 

to 

XfH 

CO 

cq 

rH 

rH 

O 




CO 

CO 

b- 

b- 

b^ 


b^ 

05 

rH 

CO 

CO 

CO 

rH 

CO 

tO 

CO 

® 



He© 

CO 


tO 

co 


b- 

b^ 

b- 

CO 

CO 

CO 

CO 

05 

05 

05 

'H 

to 



W 

CN 

CO 

oi 

• 

• 

O 

co 

• 

05 

• 

o 

• 

rH 

• 

cq 

• 

CO 

• 

• 

to 

• 

CO 

• 

• 

CO 

• 

05 

• 




rH 

rH 

rH 

rH 

rH 

05 

00 

CO 

b^ 

CO 

to 

Hfi 

CO 

cq 

rH 

f-4 

® 





b- 

CO 

CO 


tH 

CO 

hH 

CO 


05 


05 


05 

b— 

to ' 




Cq 

CO 

00 

05 

rH 

05 


o 

to 

r-i 

CO 

cq 

-t- 

co 

CO 

CO 

H 





CN 

• 

t-H 

• 

O 

• 

O 

Cq 

• 

• 

CO 

• 

b- 

• 

05 

• 

o 

• 

cq 

• 

co 

to 

• 

CO 

cq 

• 

CO 

• 





rH 

rH 

rH 

rH 

05 

oo 

b- 

co 

tO 

to 


cb 

cq 

rH 

rH 

O 






05 

O 

tO 

CO 

rH 

05 

CO 


cq 

o 

co 

CO 


CO 

cq 





H» 

O 

CO 

o 

rH 

cq 

cq 

CO 


to 

CO 

co 


CO 

oo 

05 





rH 

• 

rH 

• 

o 

to 

• 

b^ 

• 

05 

• 

rH 

• 

CO 

• 

to 

• 

■b- 

• 

05 

• 

rH 

• 

CO 

• 

to 

• 

rH 

• 

b- 

• 






rH 

rH 

05 

CO 

b- 

b- 

CO 

to 


CO 

CO 

cq 

r—i 

rH 

® 







O 

rH 

cq 

co 

CO 


00 

»o 

CO 


00 

05 

05 

05 







rH 

b- 

co 

05 

to 

rH 

b- 

CO 

05 

to 

rH 


® 

co 






rH 

vO 

CO 

rH 

CO 

CO 

05 

rH 


CO 

05 

cq 

HH 

pH 

b- 







05 

oo 

00 

b- 

cb 

tb 

tb 


CO 

cq 

cq 

rH 

rH 

® 









rH 


CO 

cq 

tO 

05 

cq 

CO 

05 

cq 

05 

CO 







1 4» 

CO 

tO 

CO 

b- 

05 

o 

rH 

co 


tO 

b- 

cq 

CO 

*s 






rH 

cq 

tp 

co 

rH 


CO 

rH 


b- 

o 

CO 

® 

co 

O 







CO 


cb 

cb 

tb 


tH 

CO 

cq 

cq 

pH 

rH 

O 

u 








o 

b- 

co 

05 

to 

cq 

CO 

tO 

o 

CO 

05 

CO 

a> 







HN 


co 

® 

CO 

CO 

o 

CO 

co 

o 

CO 


CO 

rQ 







rH 

05 

• 

co 

• 

b- 

• 

o 

• 

CO 

• 

rH 

• 

tO 

♦ 

05 

cq 

• 

05 

• 

co 

a 








CO 

co 

to 

tb 


CO 

CO 

cq 

rH 

rH 

O 

O 

p 









00 

b^ 

co 

to 

hH 


tO 

cq 

rH 

® 

® 

ri 








tdpo 

o 

cq 


CO 

CO 

o 

oq 


CO 

b- 

CO 

M 









CO 

cq 

CO 

o 


05 

co 


rH 

cc 

to 










• 

• 

• 

• 

• 

• 

• 

• 

• 


• 

0) 









to 

tO 



CO 

cq 

cq 

rH 

pH 

® 

® 

rd 










cq 


CO 

co 

o 

cq 

hH 

CO 

cq 

CO 

H> 










tO 

cq 

05 

CO 


rH 

CO 

to 

05 

cq 










l-H 

b^ 

cq 

CO 

rH 

co 

rH 

tp 

O 

b- 

to 

is 












CO 

cb 

cq 

cq 

pH 

rH 

O 

® 












cq 

CO 

o 

tO 

rH 

tO 

® 

cq 

to 

• 











o 

cq 

to 


O 

cq 

tO 

pH 

b- 

(ft 










rH 

CO 

• 

CO 

• 

CO 

• 

CO 

• 

05 

• 

• 

05 

b- 

• 

• 

O 











CO 

CO 

cq 

cq 

rH 

rH 

® 

O 

O 

H> 












CO 


cq 

O 


to 

co 

cq 













to 

CO 

rH 

05 

CO 


CO 

cq 












“H 

05 

to 

rH 

CO 

cq 

co 

CO 

'H 

h> 












cq 

cq 

• 

T* 

rH 

• 

rH 

• 

® 

• 

® 

• 

O 














b- 

CO 

i>- 

oo 

co 

CO 

05 

d 













rH 


b- 

o 

co 

to 

co 

o 












«^D 

cq 

• 

CO 

• 

• 

rH 

• 

b- 

• 

tp 

co 

• 














cq 

rH 

rH 

rH 

® 

® 

o 

CO 

. rH 















tO 

O 

co 


CO • 















00 

co 

to 

CO 

b- 

rH 

rd 

H* 













** 

tO 

• 

rH 

cq 

rH 

05 

• 

o 

CO 

• 

® 

• 

® 

CO 

t 

® 

O 













>— 






- ~ 

d 















CO 

cq 

QO 

CO 


0> 















tO 

05 

cq 

05 

CO 

tn 















o 

b- 

to 

CO 

cq 
















• 

rH 

• 

o 

• 

® 

® 

® 

O 
















hH 

cq 

CO 

r—1 

J3 
















CO 

cq 

rH 

rH 


• 

i-a, 















O 

'H 

CO 

cq 


o o o 

b- b- CO 
rH CO tO 
Cq H 

o o 


CO 

o 


cq co 
co o 


o o 


<0 

<? 

o 




















































Weight of Flat Rolled Iron 


© 

c 

fc 

u 

« 

A 


Weight of Flat Rolled Iron per foot. 




CO 

03 

05 

•O 

rH 

05 

so 

03 

O 

tO 

rH 

X^ 

rH 

o 

X^ 

co 

rH 

05 

rH 

— "7 
to 


<£> 

1> 

co 

CO 

to 

O 

rH 

co 

rH 

05 

co 

CO 

03 

X^ 

03 

SO 

03 

05 

SO 

O 

co 


X- 

4h 

03 

05 

CO 

CO 

03 

O 

tb 

tb 

03 

O 

lb 

tb 

03 

O 

tO 

CD 

CO 

to 



rH 

rH 

rH 

CO 

CO 

CO 

CO 

co 

03 

03 

03 

03 

rH 

rH 

rH 

rH 

X- 

tb 

CO 

03 



to 

03 

05 

CO 

co 

rH 

CO 

to 

03 

05 

SO 

CO 

O 

X^ 

rH 

SO 

X^ 

CO 

CO 

05 



05 

CO 

O 

co 

rH 

O 

to 

rH 

X*— 

03 

co 


O 

to 

r—< 

rH 

co 

to 

rH 

03 


tO 

jO 

cb 

rH 

CO 

sb 

rH 

r-H 

05 

SO 

rH 

rH 

05 

xb 

rf 

03 

X- 

03 

cc 

SO 

rH 



rH 

rH 

rH 

co 

Ct;? 

co 

co 

03 

03 

03 

03 

r-H 

rH 

rH 

T—1 

05 

X- 

4h 

cb 

03 



rH 

03 

05 

X- 

to 

CO 

O 

CC 

to 

CO 

rH 

05 

SO 

rH 

T—1 

rH 

o 

x^ 

tO 

CO 



rH 

CO 

rH 

t-H 

CO 

tO 

03 

co 

to 

03 

05 

tO 

03 

05 

SO 

05 

X- 

rH 

CO 

03 


tO 

4* 

rH 

05 

lb 

rH 

03 

O 

lb 

tb 

cb 

O 

cb 

SO 

cb 

rH 

03 

C5 

SO 

rH 

CO 



rH 

rH 

CO 

CO 

co 

co 

CO 

03 

03 

03 

03 

T—U 

rH 

rH 

rH 

05 

SO 

4h 

cb 

03 



co 

03 

o 

CO 

CO 

to 

CO 

rH 

05 

CO 

SO 

rH 

03 

T—H 

05 

H 

co 

so 

x^ 

00 



rH 

05 

x^ 

rH 

03 

o 

CO 

so 

CO 

rH 

05 

X- 

tO 

CO 

O 

X- 

to 

CO 

03 

rH 


lO 

03 

05 

ib 

tO 

CO 

r-H 

CO 

SO 

rH 

03 

05 

lb 

tb 

cb 

rH 

co 

so 

rH 

CO 

03 



rH 

CO 

co 

co 

co 

CO 

03 

03 

03 

03 

rH 

rH 

rH 

rH 

rH 

co 

sb 

rH 

cb 

03 

q 


CO 

03 

o 

05 

CO 

J>- 

SO 

tO 

co 

03 

rH 

O 

CO 

1>- 

SO 

05 

so 

rH 

CO 

03 

a 

to 

r—i 

o 

05 

X^ 

co 

to 

rH 

co 

03 

T—1 

O 

05 

X^ 

SO 

tO 

rH 

co 

03 

SO 

t-H 

a 

O 

cb 

tb 

cb 

rH 

05 

X^ 

tb 

cb 

• 

t-H 

05 

SO 

• 

rH 

03 

O 

rH 

co 

• 

03 

rH 



rH 

CO 

co 

co 

CO 

03 

03 

03 

©3 

03 

rH 

rH 

rH 

rH 

rH 

b 

SO 

4h 

CO 

03 

O 


03 

03 

rH 

o 

05 

05 

CO 

co 

X- 

X- 

SO 

tO 

rH 

rH 

CO 

so 

05 

co 

05 

SO 

4h 

rH 

r-H 

rH 

rH 

O 

o 

o 

o 

O 

o 

o 

O 

O 

O 

O 

03 

rH 

rH 

o 

o 

© 

ri 

QO 

cb 

4h 

03 

O 

cb 

cb 

4h 

03 

• 

o 

cb 

SO 

4h 

03 

o 

O 

O 

o 

o 

o 


CO 

CO 

CO 

CO 

CO 

03 

03 

03 

03 

03 

rH 

t-H 

rH 

rH 

rH 

co 

SO 

4h 

CO 

03 

»-*-< 


03 

03 

r-H 

rH 

r-H 

rH 

rH 

rH 

t-H 

r-H 

rH 

t-H 

rH 

r—l 

X^ 

rH 

CO 

03 

rH 

T—1 


r^N 

rH 

rH 

03 

CO 

rH 

tO 

co 

X- 

CO 

05 

O 

rH 

03 

CO 

rH 

o 

O 

o 

O 

to 

CD 

a 

CO 

4h 

03 

O 

cb 

cb 

4h 

03 

o 

05 

lb 

tO 

cb 

• 

7-H 

to 

• 

so 

x^ 

• 

cc 

Cjp 

05 

• 



CO 

CO 

co 

co 

03 

03 

03 

03 

03 

rH 

r—l 

rH 

rH 

t-H 

05 

lb 

to 

cb 

03 

T—1 

rd 


r-H 

03 

03 

03 

03 

CO 

co 

rH 

rH 

tO 

tO 

SO 

SO 

X- 

to 

rH 

so 

rH 

co 

to 

-4-J 


rH 

CO 

tO 

X— 

05 

rH 

co 

tO 

X- 

05 

rH 

CO 

to 

X- 

x^ 

CO 

QO 

05 

05 

05 

h3 

c3 

a> 

r TT 

r* 

4* 

03 

o 

cb 

cb 

• 

tO 

cb 

rH 

05 

xb 

3d 

4h 

03 

o 

05 

^H 

• 

CO 

to 

• 

SO 

• 

X- 


CO 

co 

co 

03 

03 

03 

03 

03 

rH 

rH 

r-H 

rH 

rH 

rH 

b 

x^ 

tb 

CO 

03 

rH 



o 

r-H 

03 

rH 

to’ 

CO 

X^ 

CO 

05 

O 

rH 

03 

CO 

rH 

tO 

05 

05 

o 

to 

O 

rO 


rH 

rH 

X- 

O 

co 

SO 

05 

03 

to 

05 

03 

tO 

CO 

rH 

rH 

to 

SO 

CO 

CO 

05 

o 

rH 

03 

• 

O 

cb 

• 

X^ 

tb 

cb 

• 

rH 

* 

o 

cb 

SO 

tb 

cb 

• 

r-H 

• 

O 

rH 

• 

■b- 

o 

• 

CO 

to 

eo 



co 

CO 

03 

03 

03 

03 

03 

03 

rH 

rH 

rH 

rH 

rH 

rH 

co 

sb 

to 

cb 

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rH 

co 


05 

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co 

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rH 

05 

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CO 

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to 

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a 

d 

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CO 

rH 

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05 

03 

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CO 

X- 

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rH 

03 

CO 

rH 

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to 

CO 

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05 

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cb 

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co 

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05 

03 

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so 

• 


CO 

03 

03 

03 

03 

03 

03 

rH 

rH 

rH 

rH 

rH 

rH 

05 

CO 

SO 

4h 

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03 

rH 


05 

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CO 

to 

co 

cc 

05 

rH 

03 

rH 

X- 

X^ 

05 

to 

rH 

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03 

CO 

SO 

rH 

es 

CO 

to 

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rH 

CO 

X- 

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to 

<3 

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co 

03 

SO 

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o 

03 

CO 

to 

so 

X- 

00 

P 

rH 

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05 

cb 

cb 

tb 

co 

03 

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05 

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05 

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03 

03 

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05 

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05 

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03 

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1^ 

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CO 

to 

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05 

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lb 

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03 

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03 

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o 

rH 

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rH 

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o 

03 

to 

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p 


03 

03 

03 

03 

03 

03 

rH 

rH 

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rH 

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r-H 

rH 

05 

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4h 

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r-H 










rH 













CO 


05 

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CO 

rH 

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CO 

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rH 

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03 

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CO 

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05 

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to 

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tb 

cb 

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05 

lb 

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CO 

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05 

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rH 

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05 

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03 

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03 

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co 

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CO 

to 

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X- 

tb 

4h 

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to 

rH 

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r-H 

rH 

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03 

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03 

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tb 

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03 

O 

CO 

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o 

O 

O 

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Ho 

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X^ 

rH 

rH 

CO 

rH 

rH 

co 

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03 

CO 

tO 

rH 

03 

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co 

so 

rH 

CO 

03 

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o 

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• 

• 

• 


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05 

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03 

05 

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05 

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CO 

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rH 

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rH 

05 

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b. 

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CO 


rH 

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05 

03 

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03 


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cb 

03 

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CO 

SO 

co 

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O 

• 

CO 

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05 

03 

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03 

03 

03 

03 

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rH 

r- 1 

r—l 

rH 

rH 

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rH 

cb 

lb 

SO 

to 

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05 

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CO 

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CO 

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C5 

CO 

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03 

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03 

03 

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03 

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CO 

03 

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CO 

03 

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r-H 


03 


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cb 


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CO 

03 

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to 

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CO 

03 

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CO 


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05 

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CO 

CO 

1 — 

03 

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CO 

03 

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CO 

03 

rH 

SO 


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o 

05 

CO 

CO 

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to 

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rH 

05 

co 

co 

03 

SO 

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to 


. 


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• 

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CO 

CO 

03 

03 

rH 

rH 

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©3 

o 

05 

X^ 

CO 

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CO 

03 

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o 


• 

• 


• 


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• 

• 

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03 

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r-H 

rH 

r-H 

rH 

r-H 

rH 

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05 

CO 

1^ 

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to 


co 

03 


rH 



etc 


rtO 

HO 

W 

ito 

HCS 

Ctfj 

H# 

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Ho 

coht 

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CtO 


H30 



©3 

03 

03 

03 

rH 

rH 

rH 

rH 

rH 

rH 

r-H 

rH 






















































Flat Rolled Iron per Foot 


0 


Weight of Elat Rolled Iron per Foot. 


© 

A 

bf) 

© 

£ 


co 


o ■ 

a a 

fe CO 

jg- 

lo Cb 

Q 

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w o 

S 


CO 

to 



rH 


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CO 


rn 

X- 

CO 

05 

CO 


rH 

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CO 

05 

cb 

rH 

cb 

rH 

• 

cq 

• 

rH 

• 

rH 

• 

to 

o 

to 

05 


CO 

CO 

00 

CO 

X- 

cq 

Cf> 

rH 

rH 

o 

O 

CO 

rH 

CO 

rH 

CO 

CO 

co 

O 

00 

to 

co 

o 

CO 

iO 

cb 

o 

rH 

rH 

rH 

rH 

05 

05 

co 

GO 

X— 

Jr- 

Jr- 

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CO 

CO 

co 

CO 

to 

to 

tO 

to 


05 

O 

CO 

o 


co 

CO 

o 

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CO 

to 

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o 

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CO 

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rH 

CO 


to 

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CO 

CO 


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to 

rH 

X- 

CO 

CO 


O 

to 

tO 


CN 

• 

• 


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• 

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O 

Jr- 

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co 

CO 

to 

CO 

o 

CO 

to 

CO 

rH 

CO 


rH 

rH 

05 

05 

CO 

co 

X- 

Jr- 

X- 

X- 

co 

CO 

CO 

co 

to 

to 

to 

tO 




CO 

CO 

05 


05 

to 

cq 

o 

Jr- 

to 

co 

rH 

co 

CO 

co 

rH 

05 

Jr- 


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to 

05 

cq 

CO 

05 

CO 

o 

X- 

co 

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X- 

-ch 

o 

X- 


rH 

X- 

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bq 

CO 

cb 

cb 

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cq 

05 

X- 

to 

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CO 

tb 

CO 

rH 

CO 

cb 



05 

c** 

co 

00 

Jr- 

Jr- 

X- 

CO 

CO 

CO 

co 

CO 

to 

to 

to 

tO 






rH 

X— 

HfH 

O 

Jr- 

to 

CO 

rH 

o 

CO 

co 


co 

rH 

05 

X- 

tO 



rt* 

Jr- 

cq 

00 


05 

Jr- 

to 

CO 

rn 

CO 

co 


cq 

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X- 

to 

CO 



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do 

T* 

05 

tb 

o 

do 

cb 

b 

<M 

05 

X- 

to 

CO 

rH 

cb 

cb 

b 




00 

CO 

Jr- 

Jr- 

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co 

CO 

CO 

CO 

to 

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to 

to 

tO 








CO 


rH 

05 

Jr- 

CO 

to 


CO 

cq 

o 

05 

CO 

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o 

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to 

05 

CO 

cq 

rH 

o 

05 

CO 

CO 

to 


CO 

cq 




to 

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CO 

rH 

Jr- 

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cb 

rH 

05 

Jr- 


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cb 

b 

bq 





00 

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CO 

CO 

CO 

CO 

to 

to 

to 

to 

to 



T* 







CO 

cq 

rH 

o 

o 

05 

CO 

Jr- 

X- 

CO 

CO 

to 

Htl 

co 

co 





ccH* 

cq 

CNJ 

cq 

cq 

cq 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

rH 

H 






bq 

do 

b 

bq 

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cb 

cb 

b 

bq 

O 

do 

cb 

b 

bq 

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Jr- 

co 

CO 

CO 

CO 

tO 

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to 

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to 





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co 

CO 

co 

CO 

co 

co 


tO 

CO 

cq 

cq 

cq 

cq 

cq 







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00 

05 

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cq 

cq 

cq 


CO 

X- 

CO 

05 

o 







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do 

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tb 

co 

rH 

05 

X- 

tb 

cb 

rH 

05 

cb 







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co 

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to 

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tO 

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co 


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CO 

co 

CO 

05 

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CO 

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CO 

CO 

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CO 

CO 

05 

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CO 

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cq 

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cb 

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CO 

CO 

co 


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05 CO 

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CO 

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05 Cq CO 

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CO CO 


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co *b 
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CO 

X- 

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CO 



CO 

to 

X- 

05 

O 

cq 

CO 

to 

CO 

co 

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05 

CO 

X— 

rH 

CO 

o 

T* 

CO 

cq 

CO 

CO 

tb 

bn 

bq 

rb 

05 

do 

cb 

bh 

cb 

rH 






CO 

co 

co 

co 

co 

co 


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CO 

to 

rH 

05 

tO 

cq 

CO 

co 

CO 

cq 

X- 

cq 

CO 

rH 

co 

cq 

rH 

05 

CO 

CO 

tO 

cb 

bq 

o 


tJH 

co 

CO 

CO 

CO 

co 

CO 

co 


cq 


CO 

CO 

o 

co 


X- 

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05 

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05 


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• 

to 

• 

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• 

to 


05 

do 

cb 

tb 


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t—H 

05 


CO 

co 

co 

CO 

co 

co 

co 

cq 



X- 


rH 

CO 

co 

co 

rH 


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o 

CO 

cq 

• 

X- 

co 

• 

05 

• 

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jb 

tb 

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bq 

rH 

05 

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CO 

co 

CO 

co 

CO 

cq 

cq 




co 

to 

X- 

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co 

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CO 

05 

to 

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CO 





• 

• 

• 






CO 


Cq 

rH 

O 

co 

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CO 

CO 

CO 

cq 

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05 


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05 

• 

• 

o 

• 

X- 

• 

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o 

05 

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CO 

co 

cq 

cq 

cq 






cq 

CO 

r_ 








co 

CO 

co 





CO 

05 

xb 

co 

• 

to 






Cq 

Cq 

Cl 

cq 







cq 

o 

05 






»«D 

Jr- 

to 

cq 






q 

CO 

tb 

b 







cq 

Cq 

cq 








C5 

co 







«N* 

co 

cq 







• 

• 







cq 


CO 








cq 

cq 









CO 








td» 

rH 








cq 

bq 









cq 


















cq 

































Weight or materials. 


287 


Weight Per Square Foot in Pounds. 


Thickness 
in inches. 

Cast Iron. 

Wrought or 
Sheet Iron. 

1 

Sheet Copper. 

Sheet Lead. 

Sheet Zinc 

1 

1 6 

2-346 

2-517 

2-890 

3-694 

2-320 


4-693 

5-035 

5-781 

7-382 

4-642 

3 

TB 

7-039 

7-552 

8-672 

11-074 

6-961 

4 

9-386 

10-070 

11-562 

14-765 

9-275 

7T 

T 6 

11-733 

12-588 

14-453 

18-456 

11.61 

t 

14-079 

15-106 

17-344 

22-148 

13-93 

7 

IB 

16-426 

17-623 

20-234 

25-839 

16-23 

* 

18-773 

20 141 

23-125 

29-530 

18-55 

T5 

21-119 

22-659 

26-016 

33-222 

20-87 

f 

23-466 

25-176 

28-906 

36-913 

23-19 

ii 

25-812 

27-694 

31-797 

40-604 

25-53 

I 

28-159 

30-211 

34-688 

44-296 

27-85 

13 

1 fi 

30-505 

32-729 

37-578 

47-987 

30-17 


32-852 

35-247 

40-469 

51-678 

32-47 

« 

35-199 

37-764 

43-359 

55-370 

34-81 

i 

37-545 

40-282 

46-250 

59-061 

37-13 

n 

42-238 

45-317 

52-031 

66-444 

41-78 


46-931 

50-352 

57-813 

73-826 

46-42 

it 

-51-625 

55-387 

63-594 

63-594 

51-04 

n 

56-317 

60-422 

69-375 

88-592 

55-48 

if 

61-011 

65-458 

75-156 

95-975 

60-35 

ii 

65-704 

70-493 

80-938 

103-358 

65.00 

u 

70-397 

75-528 

86-719 

110-740 

69-61 

2 

75-090 

80-563 

92:500 

118-128 

74-25 


Weight of Copper Rods or Bolts per Footj 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter 

Weight. 

Inches. 

Pounds. 

Inches 

Pounds. 

Inches. 

Pounds. 

Inches. 

Pounds. 

4 

0-1892 

l 

3-0270 

if 

10-642 

3f 

34-487 

TB 

0-2956 

Vs 

3-4170 

2 

12-108 

34 

37-081 

f 

0-4256 

n 

3-8912 

24 

13-668 

3f 

39-737 

T 

FB 

0-5794 

1 3 

1 6 

4-2688 

24 

15-325 

34 

42.568 

4 

0-7567 

*4 

4-7298 

2f 

17-075 

34 

45-455 

9 

TB 

0-9578 

1 * 
L T6 

5-2140 

24 

18-916 

4 

48-433 

a 

1-1824 

if 

5-7228 

2§ 

20-856 

44 

53-550 

11 
] 6 

1-4307 

h\ 

6-2547 

2f 

22-891 

44 

61-321 

1 

1-7027 

n 

6-8109 

2f 

25-019 

44 

68-312 

i a 

1 B 

1-9982 

l 9 

i T6 

7-3898 

3 

27-243 

5 

76-130 

i 

2-3176 

if 

7-9931 

34 

29-559 

54 

91-550 

1 5 

16 

2-6605 

u 

9-2702 

34 

31-972 

6 

109- 








_ 





































288 American Wire Gauge. 


Gauge 

num. 

Size 

inches. 

llolled Plates. 

Weight per square foot. 

Drawn Wire. 

Weight per 1000 feet. 

No. 

In. 

Iron. 

Steel. 

Copper 

Brass. 

Iron. 

Steel. 

Copper 

Brass 



Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

libs. 

Lbs. 

0000 

.4600 

18.75 

18.97 

21.36 

20.84 

566.3 

571.7 

646.8 

634.1 

000 

.4096 

16.70 

16.90 

19.01 

18.56 

449.1 

453.3 

512.9 

502.9 

00 

.3618 

14.87 

15.05 

16.93 

16.52 

356.1 

359.5 

406.8 

398.8 

0 

.3249 

13.24 

13.40 

15.08 

14.72 

282.4 

285.1 

322.5 

316.3 

1 

.2893 

11.79 

11.93 

13.43 

13.11 

224.5 

226.1 

255.8 

250.8 

2 

.2576 

10.50 

10.63 

11.96 

11.67 

177.6 

179.3 

202.9 

198.9 

3 

.2294 

9.354 

9.464 

10.65 

10.39 

140.8 

142.2 

160.8 

157.8 

4 

.2043 

8.330 

8.428 

9.486 

9.255 

111.7 

112.7 

127.5 

125.1 

5 

.1819 

7.418 

7.505 

8.448 

8.242 

88.59 

89.43 

101.2 

99.2.) 

G 

.1620 

6.606 

6.683 

7.523 

7.340 

70.26 

70.92 

80.25 

78.67 

7 

.1443 

5.882 

5.952 

6.699 

6.536 

55.7 L 

56.24 

63.64 

61.38 

8 

.1285 

5.238 

5.300 

5.966 

5.821 

44.18 

44.60 

50.46 

49.48 

9 

.1144 

4.665 

4.720 

5.313 

5.184 

35.04 

35.37 

40.02 

39.24 

10 

.1019 

4.154 

4.203 

4.731 

4.616 

2S.26 

28.05 

31.73 

31.11 

11 

.0907 

3.700 

3.743 

4.213 

4.110 

22.03 

22.24 

25.16 

24.6S 

1‘2 

.0808 

3.294 

3.333 

3.752 

3.661 

17.47 

17.64 

19.95 

19.57 

13 

.0720 

2.934 

2.96S 

3.341 

3.260 

13.85 

13.99 

15.82 

15.52 

14 

.0641 

2.613 

2.643 

2.978 

2.903 

10.99 

11.09 

12.55 

12.31 

15 

.0571 

2.327 

2.354 

2.650 

2.585 

8.717 

8.899 

9.953 

9.761 

16 

.0508 

2.072 

2.096 

2.359 

2.302 

6.913 

6.978 

7.896 

7.741 

17 

.0152 

1.845 

1.867 

2.101 

2.050 

5.481 

5.532 

6.26 L 

6.137 

18 

.0403 

1.643 

1.662 

1.872 

1.826 

4.347 

4.387 

4.965 

4.867 

19 

.0359 

1.463 

1.480 

1.666 

1.626 

3.447 

3.479 

3.937 

3.861 

20 

.0320 

1.303 

1.318 

1.484 

1.448 

2.735 

2.761 

• 3.125 

3.064 

21 

.0285 

1.160 

1.174 

1.321 

1.289 

2.168 

2.188 

2.476 

2.428 

22 

.0253 

1.033 

1.045 

1.176 

1.148 

1.720 

1.736 

1.964 

1.926 

23 

.0226 

.9203 

.9310 

1.048 

1.023 

1.363 

1.376 

1.557 

1.527 

24 

.0201 

.8195 

.8291 

.9334 

.9105 

1.081 

1.091 

1 .'235 

1.211 

25 

.0179 

.7298 

.7383 

.8311 

.8109 

.8575 

.8656 

.9795 

.9603 

26 

.0159 

.6499 

.6575 

.7401 

.7221 

.6801 

.6864 

.7768 

.7616 

27 

.0142 

.5787 

.5855 

.6591 

.6430 

.5393 

.5444 

.6160 

.6039 

28 

.0126 

.5154 

.5214 

.5869 

.5726 

.4277 

.4317 

.4885 

.4789 

29 

.0113 

.4580 

.4643 

.5227 

.5099 

.3391 

.3422 

.3873 

.3797 

30 

.0100 

.4087 

.4135 

.4654 

.4541 

.3699 

.2714 

.3072 

.3012 

31 

.0089 

.3640 

.3683 

.4145 

.4044 

.2134 

.2153 

.2437 

.2389 

32 

.0080 

.3241 

.3279 

.3691 

.3601 

.1691 

.1707 

.1932 

.189 4 

33 

.0071 

.2887 

.2920 

.3287 

.3207 

.1341 

.1354 

.1532 

.1502 

34 

.0063 

.2570 

.2600 

.2927 

.2856 

.1063 

.1073 

.1216 

.1192 

35 

.0056 

.2289 

.2316 

.2606 

.2543 

.0845 

.0853 

.0965 

.0947 

3G 

.0050 

.2039 

.2062 

.2322 

.2265 

.0669 

.0675 

.0764 

.0750 

37 

.0045 

.1816 

.1837 

.2067 

.2017 

.0531 

.0536 

.0606 

.059 4 

38 

.0040 

.1617 

.1636 

.1841 

.1796 

.0118 

.0424 

.0480 

.0471 

39 

.0035 

.1440 

.1456 

.16 tO 

.1600 

.0334 

.0337 

.0381 

.0374 

40 

.0031 

.1282 

.1297 

.1160 

.1424 

.0268 

.0267 

.0302 

.0297 

Spec 

. grav. 

7.828 

7.92 

8.917 

8.70 

7.85 

7.93 

8.96 

8.78 


The American Wire Gauge is introduced and manufactured by J. R. Brown & 
Sharpe, of Providence, It. I., and is to be had in the principal hardware stores in the 
country. It is adopted by most manufacturers of plates and wire, and is now con¬ 
sidered-the American Standard Gauge. 
































Birmingham Gauge. 


289 


Birmingham Gauge for Wire, Sheet Iron and Steel. 


Thiokuess by 

Thickness in 

Weight per Square Foot in Pounds. 

Sheet and Sheet Cast Sheet .. 

Thickness in 

the gauge. 

inches. 

Boiler Iron. 

Steel. 

Copper. 


inches. 

No. 0000 

0.454 

18.207 

18.259 

20.566 

26.75 

7:16 

000 

0.425 

17.053 

17.280 

19.252 

25.06 

27:64 

00 

0.380 

15.247 

15.451 

17.214 

22.42 

3:8 

0 

0.340 

13.7 

J4.0 

15.6 

20.06 

11:32 

1 

0.300 

12.1 

12.4 

13.8 

17.72 

5:16 

2 

0.284 

11.4 

11.7 

13.0 

16.75 

9:32 

3 

0.259 

10.4 

10.6 

11.9 

15.26 

1:4 

4 

0.238 

9.60 

9.80 

11.0 

14.02 

7:32 

5 

0.220 

8.85 

9.02 

10.1 

12.98 

7:32 

6 

0.203 

8.17 

8.33 

9.32 

11.98 

7:32 

7 

0.180 

7.24 

7.38 

8.25 

10.63 

3:16 

8 

0.165 

6.65 

6.78 

7.59 

9.73 

3:16 

9 

0.148 

5.96 

6.08 

6.80 

8.72 

5:32 

10 

0.134 

5.40 

5.51 

6.16 

7.90 

5:32 

11 

0.120 

4.83 

4.93 

5.51 

7.08 

1:8 

12 

0.109 

4.40 

4.50 

5.02 

6.42 

1:8 

13 

0.095 

3.83 

3.91 

4.37 

5.60 

3:32 

14 

0.083 

3.34 

3.41 

3.81 

4.90 

3:32 

15 

0.072 

2.90 

2.96 

3.31 

4.25 

1:16 

1G 

0.065 

2.62 

2.67 

3.00 

3.83 

1:16 

17 

0.058 

2.34 

2.39 

2.67 

3.42 

1:16 

18 

0.049 

1.97 

2.01 

2.25 

2.90 

1:16 

19 

0.042 

1.69 

1.72 

1.93 

2.48 

3:64 

20 

0.035 

1.41 

1.42 

1.61 

2.04 

3:64 

21 

0.032 

1.29 

1.31 

1.47 

1.89 

3:64 

22 

0.028 

1.13 

1.15 

1.29 

1.65 

1:32 

♦ 23 

0.025 

1.00 

1.02 

1.14 

1.47 

1:32 

24 

0.022 

0.885 

0.903 

1.01 

1.30 

1:32 

25 

0.020 

0.805 

0.820 

0.918 

1.18 

1:32 

26 

0.018 

0.724 

0.738 

0.826 

1.06 

1:64 

27 

0.016 

0.644 

0.657 

0.735 

0.94o 

1:64 

28 

0.014 

0.563 

0.574 

0.642 

0.826 


29 

0.013 

0.523 

0.533 

0.597 

0.767 


30 

0.012 

0.483 

0.493 

0.551 

0.708 


31 

0.010 

0.402 

0.410 

0.480 

0.600 


32 

0.009 

0.362 

0.370 

0.420 

0.532 


33 

0.008 

0.322 

0.328 

0.370 

0.472 


34 

0.007 

0.282 

0.288 

0.323 

0.413 


35 

0.005 

0.230 

0.235 

0.262 

0.309 


36 

0.004 

0.170 

0.173 

0.194 

0.236 



Birmingham Gauge for Silver and Gold. 


No. 

Thick. 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick 

Inch. 

1 

.004 

7 

.015 

13 

.036 

19 

.064 

25 

.095 

31 

.133 

2 

.005 

' 8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

.143 

3 

.008 

9 

.019 

15 

.047 

21 

.072 

27 

.113 

33 

.145 

4 

.010 

10 

.024 

16 

.051 

22 

.074 

28 

.120 

34 

.148 

5 

.013 

11 

.029 

17 

.057 

23 

.077 

29 

.120 

35 

.158 

6 

.013 

12 

.034 

18 

.061 

24 

.082 

30 

.126 

38 

.167 


19 





























290 Proportion op Bolts and Nuts._ 


Proportion of Bolts and Nnts. Munaber of Threads per In. 









Diameter. 

Number of 
per Ii 

Threads 

ich. 

jff 

0 

rf 


□ 

0 

sf§ 

jj 

fjj 

gj 

3 inch. 

Ui 

5f 

5 

7 t V 

21 

i 

20 

10 

2| 

4f 

4f 

41 

6f 

21 

T6 

18 

9 

21 

3f 

4f 

4f 

ty.3 

6 

2 

3 

8 

16 

9 

21 

3f 

3f 

3f 

5j6 

If 

tV 

: 14 

8 

2 

3 

31 

3f 

4f 

If 

1 

2 

12 

7 1 

n 

2f 

3f 

3 

4& 

11 

f 

11 

7 

if 

2f 

3 

2f 

3f 

If 

3 

¥ 

10 

6 

if 

2f 

2f 

2f 

311 

11 

7 

8 

9 

6 

if 

! 2f 

2f 

21 

31 

If 

1 

8 

5 

«it 

o 3 

2 

2f 

21 

3f 3 6 

ItV 

If 

7 

4 ! 

« 11 

If 

2f 

2 

2f 

1 

If 

7 

31 i 

1 i 

If 

If 

If 

2f ' 

1 

11 

6 

3 | 

2 1 

a 

If 

] 

If 

If 

2j\ 

7 

8 

If 

5 

2i; 

s i 

ItV 


11 

If 

2f 

f 

2 

41 

21 j 

3 

¥ ^ 

HV 


If 

1^ 

If 

5. 

8 

21 

4 

2 ; 

f 

1 


If 

Her 

HI 

1 r 

2 

3 

31 

* 

T¥ 

T 

~5 


L 

1 

HV 

1 

T 

31 

31 

3 


• f 


f 

| 

H 

T¥ 

4 

3 



3 

¥ 


7 

■S' 

3 

¥ 

He 

3 

■g: 

41 

2f 

i 

3 

8 

tV 


f 

3 

¥ 

HV 

T¥ 

5 

2f 


5 

TU 

TS 


I 

TS 

11 

1 6 

tV 

^1 

2f 


1 

¥ 

3 

8 

7 

TS 

TS 

1 1 

T¥ 

1 

6 

21 

■ 


I j 

The above proportions of bolts and nuts are established by Sir Joseph Whitworth. 


Weight in Pounds of lVnt and Bolt-Head. 


Diameter of Bolt in Inches. 


Head and Nut. 

1 

¥ 

3 

¥ 

1 

5 

J 

2 

¥ 

f 

1 


11 

If 

2 

21 

3 j 

Hexagon, 

.017 

.05T 

.128 

.267 

.43 

.73 

1.1 

2.14 

3.77 

562 

8.75 

17.2 

28.8 

Square, . . 

.021 

.070 

.164 

.321 

.553 

.882 

1.31 

2.56 

4.42 

7.00 

10.5 

21. 

36.4 
















































Screw-Threads. 


291 


Proportion of Screw-Threads, Nuts and Boltlieads. 


Diam.of Threads 
Screw per inch. 


Diamet. 

of core. 


Width 
of flat. 



• 1/1 

■5/16 

•3/8 

•7/16 

• 1/2 

•9/16 

•5/8 

•3/4 

•7/8 

1* 

1 - 1/8 

1-1/4 

1-3/8 

1 - 1/2 

1*5/8 

1-3/4 

1- 7/8 

2- 

2-1/4 

21/2 

2- 3/4 

3- 

3-1/4 

3-1/2 

3- 3/4 

4- 

4-1/4 

4T/2 

4- 3/4 
. 5- 

5- 1/4 

5-1/2 

5- 3/4 

6 - 


20 

18 

16 

14 

13 

12 

11 

10 

9 

8 

7 

7 

6 

6 

5-1/2 

5- 

5- 

4-1/2 

4-1/2 

4- 

4- 

3-1/2 

3-1/2 

3-1/4 

3- 

3- 

2-7/8 

2-3/4 

2-5/8 

2 - 1/2 

2 - 1/2 

2-3/8 

2-3/8 

2-1/4 


n85 

•240 
•294 
•344 
•400 
> -454 
-.507 
•620 
•731 
•837 
•940 
1-065 
1-160 
1-284 
1-389 
1-490 
1-615 
1-712 

1- 962 

2- 175 
2-425 
2-628 

2- 878 

3- 100 

3-317 

3-566 

3- 798 

4- 027 

4-255 

4-480-- 

4-730 

4- 953 

5- 203 

5-423 


•0062 

•0070 

•0078 

•0089 

•0096 

•0104 

•0113 

•0125 

•0140 

•0156 

•0180 

•0180 

•0210 

•0210 

•0227 

•0250 

•0250 

•0280 

•0280 

•0310 

•0310 

•0357 

•0357 

•0384 

•0410 

•0410 

•0435 

•0460 

•0480 

•050d 

•0500 

•0526 

•0526 

•0555 


1 Outside 

Inside 

Diagonal. 

diamet. 

i 

diamet. 




•9/16 

•1/2 

•11/16 

•11/16 

•19/32 

•13/16 

•25 f 32 

•11/16 

•31/32 

•43/48 

•25/32 

1-1/16 

1- 

-7/8 

1-1/4 

1-7/64 

•31/32 

1-5/16 

1-7/32 

"1*1/16 

1-1/2 

T7/16 

1-1/4 

1-3/4 

1-21/32 

1-7/16 

2-1/32 

1-7/8 

1-5/8 

2-5/16 

2-3/32 

1-13/16 

2-1/2 

2-5/16 

2- 

2-27/32 

2-1/2 

2-3/16 

3-1/10 

2-3/4 

2-3/8 

3-3/8 

2-15/16 

2-9/16 

3-5/3 

3-3/16 

2-3/4 

3-29/32 

3-13/32 

2-15/16 

4-3/16 

3-5/8 

3-1/8 

4-7/16 

4-1/16 

3-1/2 

4-31/32 

4-1/2 

3-7/8 

5-1/2 

4-29/32 

4-1/4 

6- 

5-3/8 

4-5/8 

6-9/16 

5-3/4 

5- 

7-1/8 

6-7/64 

5-3/8 

7-5/8 

6-5/8 

5-3/4 

8-3/16 

7-3/64 

6-1/8 

8-11/16 

7-1/2 

6-1/2 

9-1/4 

7-31/32 

6-7/8 

9-3/4 

8-3/8 

7-1/4 

10-9/32 

8-13/16 

7-5/8 

10-13/16 

9-1/4 

8- 

11-3/8 

9-11/16 

8-3/8 

11-20/32 

10-1/8 

8-3/4 

12-7/16 

10-9/16 

9-1/8 

12-9/10 


height 
of h’d. 



• 1/4 
•19/64 
•11/32 
•25/64 : 

• 7/16 
•31 /64 
•17/32 

• 5/8 
•23/32 
*13/16 
•29/64 

1- 

1-3/32 

13/16 

1-9/32 

1 - 8/8 

1-15/32 

1-9/16 

1-3/4 

1- 15/16 

2 - 1/8 
2-5/16 
2 - 1/2 
2-11/16 

2- 7/8 

3- 1/16 
3-1/4 
3-7/16 
3-5/8 

3- 13/16 

4- 

4-3/16 

4-3/8 

4-9/16 


Mr. Whitworth makes the angle of the threads 55°, with round top and bot¬ 
tom ; whilst William Sellers makes the angle of the thread 60°, with Hat top and 
bottom, and of the following proportions, which are recommended by a special 
committee appointed by the Franklin Institute of Philadelphia. For full infor¬ 
mation see Journal of the Institute, May, 1864, and Jan., 1865. 

Notation of letters. All dimensions in inches. 


D — outside diameter of screw, 
d =£= diameter of root of thread, or of 
hole in the nut. 
p = pitch of screw. 
t = number of threads per inch. 
f — flat top and bottom. 
o = outside diameter of hexagon nut 
or bolthead. 


P = 


V 16 D + 10 — 2-909 


16-64 


i = inside diameter of hexagon, or 
side of square nut or bolthead. 
s = diagonal of square nut or bolthead. 
h = height of rough or unfinished bolt- 
head. 

The height of finished nut or bolt- 
head is made equal to the diameter D 
of the screw.* 

t = — • s = 1-414 i. 

P 


d = D — 


1-299 


3 D 1 
~ 2 ~ + 8 


o = 1-155 i. 


/ = 


t ' 2 ' 8 ' ^ 8 

* Whitworth makes the height of the nut about half the hexagon diameter o. 

































Gkariwo. 



2C2 

GEARING. 


Letters denote. 

P = pitch,—the distances between the centres of two teeth in thft 
pitch circle. 

D = diameter 
C = circumference 
M — number of teeth 
N = number of revolutions ) 
d — diameter -j 

c = circumference f 

m = number of teeth ( of the P imon * 

n = number of revolutions ' 


.of the wheel. 


Pitch 


r = 


p= 


c 

M 

it D 


No. of teeth 


M 


M = 


n. D 



r C= PM 

- 5 

Circum. < 



1 

[0=TtD 

- 6 


,d- pm 

Tt 

- 7 

Diameter - 

ii 


j 

- 8 


Formula 2. Pitch P^ 


Example 1. A wheel of D = 40 inches in diameter, is to have M = 
75 teeth. Required the pitch P = ? 

3*14X40 , 1 

---= 1*66 inches nearly. 

75 

I Example 2. The pitch of teeth in a wheel, is to be P = 1*71 inches, and 
having M — 48 teeth. Required the diameter D = ? of the wheel. 

Formula 7. Diam. D — - = 26*14 in. of the piten circle 

3*14 r 
























CfEARINQ. 


2C3 


Construction of Teeth for Wheels* 

Draw the radius R r, and pitch circle P P. Through r draw the line o of at 
an angle of 75° to the radius R r. 


Half the angle be 
tween two teeth in the ] 


f wheel, v = • 

-I # 


'•pinion, V — 


180 

m 


Diameter of the 


D : d = sin. V : sin. v. 

d sin. V 
sin. v 

D sin. v 


wheel, D '■ 


pinion, d = 


sin. V 

Pitch (chord) of teeth f wheel, P = D sin. v. 
in the pitch circle 1 pinion, P= d sin. V. 

Approximate pitch in the wheel P = 0’028 D. ■ 


( wheel, M = 

d M 


Number of teeth about •< 

pinion, m = ^ 

Thickness of tooth, a ’== 0*46 P* - 
Bottom clearance, b = 0*4 P. 

Depth to pitch line, c =* 0*3 P. 

Distance r o f d = 

1 2 (m — 11 ) 


Distance r o’, e = 0T1 P 1/ 


m 


10 

11 

12 

13 

14 


* If a wheel of more than 80 teeth is to gear a pinion of less than 20 teeth, 
and the wheel and pinion are of the same kind of materials; take the thickness 

/• wheel, a = P ^0’42 + - * 15 

of the tooth in the •< 

( pinion, a = 0’5 P^l — 16 

A rask is to he considered as a wheel of 200 teeth. 














294 


Gearing. 


Example with Plate I < 

Example. A wheel of D — 48 inches diameter is to gear a pinion about 8 
revolutions to 1. Required a complete construction of the gearing? 

7 

8 


Approximate pitch P — 0‘028X48 = 1 , 34 in. - 
wheel, M'— 


3*1^<48 = n2 


Number of teeth 
in the 


1*34 
112 


pinion, m = 


= 14 


Half the an- ( 


wheel, v = 


180 


gle between •< 

two teeth in j . . ^ 

fho '-pinion V = 


112 

180 


1°36\ sm=0*028. 


the 

s? •* « ® 

Diameter of pinion 




=12o51\ sin= 0-2224. 
= 6*043 in. 


14 

48X0-028 


0-2224 


Draw the pitch circle for the wheel and pinion so that they tangent one an 
other at r on a straight line between the centres of the circles. 

Pitch in the gearing P = 48X0*028=1*344 in. - 5 

Take this chordial pitch in a pair of compasses, and set it off in the pitch 
circles. 


Thickness of 
tooth 


wheel a ~ 1*344 ^0*42 + \v=0"592in. 

V 700 / 


15 


pinion 0-5 x <l*344^1-^Q^=0-645in. 16 

Set off the thickness of tooth in the corresponding pitch circles. 

Bottom clearance b = 0-4X1-344 = 0*5376 in. - 11 

Depth to pitch line c = 0*3 < 1*344=0*4032 in. - -12 

e =* 0-11X1-344 s/Ti2 = 0*7126 in. 14 

Set off these distances on the line o o' from r,—d beyond and e within the 
pitch circle for the wheel; then o is the centre and o m radius for the flank m. 
o' the centre and o' n radius for the face n. Draw circles through o and o' con¬ 
centric with the pitch circle of the wheel. 


14 




Distances r o and f 
r o' in the wheel j 


13 


Distances r o and 
r o' in the pinion 


' = n-i 


e = 0*11X1-344^/14 = 0-356 in. 

Proceed with the pinion similar as the wheel 

On the plate is a scale of inches and decimals, which will be 
venient for the above measurements. 


13 

14 

eon 





















OS ^ CM N 


J WjSfVstrom . 


f/ate y. 































































































. 




. 




















‘ • 
































































- 






























































Strength op Teeth. 


295 


Oil tlie Strength, of Teeth in Cast-iron Wheels. 

P = pitch, a = thickness, and h = face of teeth in inches. 

S = strain in pounds, H = horsepower, and V— velocity in 
feet per second in the pitch circle. 

Pitch P. 

P= —— 

460 h 

p _1*2 H 

h V ' 

Thickness a. 

s 

a ~ 1000 h 

0.55 H 

a =- . 

h V 

Face h. 

h~ S 

1000 a * 

, 1.2 H 

h =--. 

PF 

Strain S. 

S= 1000 ah. 

S = 460 P h. 

Horsepower H. 

rr h a V 

H - 0.55 : 

H- hPV . 

1.2 

Velocity V. 

0.55 H 

h a 

V 12 * 
hP ‘ 


The face h is generally made = 2.5 P = 5.435 a. 

a = 0.46 P, and P = 2.1739 a. 


To Find the Diameter of Axles and Shaft. 

Letters denote , 

d = diameter in inches, in the bearing; and the length of the bearing 1.5(Z. 
>F = weight in pounds, acting in the bearing. 


Water¬ 

wheels. 


d = 


Vw~, 

of cast iron. 

Common i 

y , y 

r d — ■—- of cast iron. 

18 

Machinery J 

| 24 

y'w r 

-— of wrought iron. 

in good I 
order. \ 

1 V w 

[ d = —- of wrought iron. 

21 


28 


Example 1. A water-wheel weighs 58,680 pounds, and is supported in two bear¬ 
ings. Required the diameter of the wheel axles? The weight acting in each 
bearing will be 58680 : 2 = 29340 pounds, and 

diameter d = ° = 8.15 inches of wrought iron. 

Example 2 Fig. 226, page 307. Required the diameter of the axle in the 
wheel when'the weights P + Q = 4864 pounds? If the wheel is supported in 
two bearings W— 4864 : 2 = 2432 pounds. 

-j/ 2432 

diameter d = -= 1.76 inches of wrought iron. 

(Continued on page 298.) 
































Standard Pitch of Gear-Wheels. 


. 29G 


STANDARD PITCH OF GEAR-WHEELS. 

The difficulty in finding cog-wheel patterns made at different establish¬ 
ments to gear correctly into one another is well known, and much time and 
money is lost for the want of a standard scale of pitch in gearings. The 
pitch of teeth in a cog-wheel should always be understood to mean the 
chord-pitch , and not the arc-pitch, because equal arc-pitch in wheels of 
widely different diameters will not gear well. 

The pitch of gear-wheels should be even measures of the inch and binary 
fractious thereof, and the number of teeth and diameter of pitch-circle should 
be regulated accordingly. 

The following pitch-table is offered or proposed as standard, in which the 
first column varies with sixteenths of an inch from 0 to 1 inch, with eighths 
from 1 to 3 inches, and with quarters from 3 to 7 inches. The pitch of wheels 
from to 7 inches should not be made of any other measure than of those 
in the table, and the fractions with the most decimals should be avoided as 
much as possible: 


Standard Pitcli for Gear-Wheels. 


Pitch from T V to 1 inch. 

Pitch from 1 to 3 inches. 

Pitch from 3 to 7 inches. 

Binary Decimals. 

1 in. Decimals. 

S in. Decimals. 

T i s = 0-0625 
| = 0'125 

1}= 1-12.5 

3f = 3-25 

If = 1-25 

3f = 3-5 

T 3 S = o-l 875 
i = 0-25 

If = 1-375 

3f = 3-75 

lf= 1-5 

4 iD. 

y B g = 0-3125 

If = 1625 

4f = 4-25 • 

£ = 0-375 

If = 1-75 

4f = 4*5 

t 7 s = 0-4375 

lf = 1-875 

4f = 4-75 

r = 0'5 

2 in. 

5 in. 

T ® 6 = 0-5625 

2f = 2.125 

5f = 5-25 

| = 0625 

2f = 2-25 

5f = 5-5 

i| = 0-6875 

2f = 2-375 

of = 5-75 

f = 0-75 

2£ = 2-5 

6 in. 

U = 0-8125 

2g = 2-625 

6f = 6-25 

| = 0-875 

2f = 2-75 

6f = 6*5 

= 0-9375 

2f = 2.875 

6f = 6-75 

1 in. 

3 in. 

7 in. 


The width of the face should be two and a half the pitch. 


The following table contains the proportions of number of teeth, diameter, 
and chord-pitch of wheels from 6 to 250 teeth. The first column is the num¬ 
ber of teeth, the second is the diameter when the chord-pitch is unit, and 
the third column is the chord-pitch when the diameter is the unit. 

Example 1.—What diameter is required for a wheel of 45 teeth and chord- 
pitch If inches? Opposite 45 teeth we find the diameter. 

14.3356 X 1.75 = 25.0874 inches in pitch-line. 

Example 2.—A wheel of 62.35 inches diameter in the pitch-line has 198 
teeth. What is the chord-pitch? 

Pitch = 62.35 X 0.015866 = 0.989 of an inch. 

That wheel will not work with the standard gear. 

The number of teeth multiplied by the pitch gives the length of the pitch- 
polygon and not the pitch-circle. The difference between the length of the 
two pitch-lines is greater the less number of teeth in the wheels. For 250 
teeth and one-inch pitch the difference is only part of an inch in the 
whole pitch-line. 

For properly-constructed ra?-gearing there should be no clearance between 
the teeth, as shown in Plate I., but the thickness a of the teeth should be 
half the pitch. 









Proportion op Gear-Wheels. 


297 


No. 

Teeth. 

Diam¬ 

eter 

when 

P = l. 

Pitch 

when 

2> = 1. 

No. 

Teeth. 

Diam¬ 

eter 

■when 

i» = l. 

Pitch 

■when 

D=rl. 

No. 

Teeth. 

Diam¬ 

eter 

when 

P—l. 

Pitch 

■when 

D = 1. 

No. 

Teeth. 

Diam¬ 

eter 

when 

P= 1. 

Pitch 

when 

D — 1. 

6 

2.0000 

.50000 

66 

21.016 

.04758 

126 

40.111 

.02493 

186 

59.208 

.01689 

7 

2.3068 

.43358 

67 

21.334 

.04687 

127 

40.429 

.02473 

187 

59.527 

.01680 

8 

2.6131 

.38268 

68 

21.652 

.04618 

128 

40.748 

.02454 

188 

59.845 

.01671 

9 

2.9238 

.34202 

69 

21.970 

.04552 

129 

41.066 

.02435 

189 

60.163 

.01662 

10 

3.2361 

.30902 

70 

22.289 

.04486 

130 

41.384 

.02416 

190 

60.482 

.01653 

11 

3.5490 

.28177 

71 

22.607 

.04423 

131 

41.702 

.02398 

191 

60.800 

.01645 

12 

3.8637 

.25882 

72 

22.925 

.04361 

132 

42.021 

.02380 

192 

61.118 

.01636 

13 

4.1785 

.23932 

73 

23.243 

.04307 

133 

42.330 

.02362 

193 

61.436 

.01628 

14 

4.4940 

.22252 

74 

23.562 

.04242 

134 

42.657 

.02344 

194 

61.755 

.01619 

15 

4.8097 

.20791 

75 

23.880 

.04187 

135 

42.976 

.02327 

195 

62.073 

.01611 

16 

5.1259 

.19509 

76 

24.198 

.04131 

136 

43.294 

.02310 

196 

62.391 

.01603 

17 

5.4423 

.18374 

77 

24.516 

.04091 

137 

43.612 

.02293 

197 

62.710 

.01595 

18 

5.7588 

.17365 

78 

24.835 

.04026 

138 

43.931 

.02276 

198 

63.028 

.01587 

19 

6.0756 

.16460 

79 

25.153 

.03976 

139 

44.250 

.02260 

199 

63.346 

.01579 

20 

6.3925 

.15643 

80 

25.471 

.03926 

140 

44.567 

.02244 

200 

63.665 

.01571 

21 

6.7095 

.14904 

81 

25.789 

.03878 

141 

44.885 

.02228 

201 

63.983 

.01563 

22 

7.0266 

.14231 

82 

26.108 

.03830 

142 

45.204 

.02212 

202 

64.301 

.01555 

23 

7.3338 

.13636 

83 

26.426 

.03784 

143 

45.522 

.02197 

203 

64.620 

.01547 

24 

7.6613 

.13053 

84 

26.744 

.03739 

144 

45.840 

.02182 

204 

64.938 

.01540 

25 

7.9787 

.12533 

85 

27.062 

.03695 

145 

46.158 

.02167 

205 

65.256 

.01532 

26 

8.2962 

.12054 

86 

27.381 

.03652 

146 

46.477 

.02152 

206 

65.575 

.01525 

27 

8.6138 

.11609 

87 

27.699 

.03611 

147 

46.795 

.02137 

207 

65.893 

.01517 

28 

8.9315 

.11196 

88 

28.017 

.03569 

148 

47.113 

.02122 

208 

66.211 

.01510 

29 

9.2493 

.10811 

89 

28.335 

.03529 

149 

47.432 

.02108 

209 

66.529 

.01503 

30 

9.5668 

.10453 

90 

28.654 

.03490 

150 

47.750 

.02094 

210 

66.848 

.01496 

31 

9.8845 

.10117 

91 

28.972 

.03452 

151 

48.068 

.02080 

211 

67.166 

.01488 

32 

10.202 

.09800 

92 

29.290 

.03414 

152 

48.386 

.02067 

212 

67.484 

.01482 

33 

10.520 

.09506 

93 

29.608 

.03377 

153 

48.705 

.02553 

213 

67.802 

.01475 

34 

10.838 

.09226 

94 

29.927 

.03341 

154 

49.023 

.02039 

214 

68.121 

.01470 

35 

11.156 

.08964 

95 

30.245 

.03306 

155 

49.341 

.02029 

215 

68.439 

.01461 

36 

11.474 

.08716 

96 

30.563 

.03272 

156 

49.660 

.02014 

216 

68.757 

.01454 

37 

11.792 

.08480 

97 

30.881 

.03238 

157 

49.978 

.02001 

217 

69.076 

.01448 

38 

12.110 

.08257 

98 

31.200 

.03205 

158 

50.296 

.01988 

218 

69.394 

.01441 

39 

12.427 

.08049 

99 

31.518 

.03173 

159 

50.614 

.01976 

219 

69.712 

.01434 

40 

12.745 

.07846 

100 

31.836 

.03141 

160 

50.933 

.01963 

220 

70.031 

.01428 

41 

13.064 

.07653 

101 

32.154 

.03100 

161 

51.251 

.01951 

221 

70.349 

.01421 

42 

13.382 

.07476 

102 

32.473 

.03079 

162 

51.569 

.01939 

222 

70.667 

.01415 

43 

13.700 

.07299 

103 

32.791 

.03049 

163 

•51.888 

.01927 

223 

70.985 

.01409 

44 

14.018 

.07134 

104 

33.109 

.03021 

164 

52.206 

.01915 

224 

71.304 

.01402 

45 

14.336 

.06976 

105 

33.427 

.02992 

165 

52.524 

.01904 

225 

71.622 

.01396 

46 

14.654 

.06826 

106 

33.745 

.02963 

166 

52.842 

.01892 

226 

71.940 

.01390 

47 

14.972 

.06679 

107 

34.064 

.02936 

167 

53.161 

.01881 

227 

72.259 

.01384 

48 

15.290 

.06540 

108 

34.382 

.02908 

168 

53.479 

.01870 

228 

72.577 

.01378 

49 

15.608 

.06407 

109 

34.700 

.02882 

169 

53.797 

.01859 

229 

72.895 

.01372 

50 

15.926 

.06279 

110 

35.018 

.02856 

170 

54.116 

.01848 

230 

73.213 

.01366 

51 

16.244 

.06156 

111 

35.337 

.02830 

171 

54.434 

.01887 

231 

73.532 

.01360 

52 

16.562 

.06038 

112 

35.655 

.02805 

172 

54.752 

.01826 

232 

73.850 

.01354 

53 

16.880 

.05925 

113 

35.973 

.02780 

173 

55.070 

.01816 

233 

74.168 

.01348 

54 

17.198 

.05815 

114 

36.292 

.02755 

174 

55.389 

.01805 

234 

74.487 

.01342 

55 

17.517 

.05709 

115 

36.610 

.02731 

175 

55.707 

.01795 

235 

74.805 

.01337 

56 

17.835 

.05607 

116 

36.928 

.02708 

176 

56.025 

.01785 

236 

75.123 

.01331 

57 

18.153 

.05509 

117 

37.246 

.02685 

177 

56.344 

.01775 

237 

75.442 

.01325 

58 

18.471 

.05414 

118 

37.565 

.02662 

178 

56.662 

.01765 

238 

75.760 

.01320 

59 

18.789 

.05322 

119 

37.883 

.02640 

179 

56.980 

.01755 

239 

76.078 

.01314 

60 

19.107 

.05234 

120 

38.201 

.02618 

180 

57.299 

.01745 

240 

76.396 

.01309 

61 

19.425 

.05148 

121 

38.520 

.02596 

181 

57.617 

.01736 

241 

76.715 

.01303 

62 

19.744 

.05065 

122 

38.838 

.02575 

182 

57.935 

.01726 

242 

77.033 

.01298 

63 

20.062 

.04982 

123 

39.156 

.02554 

183 

58.253 

.01717 

243 

77.351 

.01293 

64 

20.380 

.04907 

124 

39.475 

.02533 

184 

58.572 

.01707 

244 

77.670 

.01287 

65 

20.698 

.04831 

125 

39.793 

.02513 

185 

58.890 

.01698 

245 

77.988 

.01282 

























298 


Strength of Materials. 


Example 3. The pressure on the steam piston, in a walking beam engine is 
26000 pounds. Required the diameter of the beam journals? 

diameter d = — = 5*64 inches the centre one. 

Zo 

d = = 4 inches at the ends. 

Zo 

In this example it is supposed that the beam is worked by a fork connecting 
rod. 



D 


V F E 


</ 


JS 

n 


D == inches wrought iron. 

E = radius of crank in feet. 

F = force from the steam piston, lbs. 


D:d= v' E 


t r 


Z> = 4 


• 3 v- 

V n 


H = horse-power transmitted. 
n = number revolutions per minute. 


When an axle or shaft not only ser ies as a fulcrum, but effect is transmitted 
by the act of twisting it, the diameteif is to be caluulated as follow. 

Example l. The pressure on the piston in a steam engine is F= 45,600 
pounds, applied direct on a crank of R = 3 feet radius. Required the diameter 
of the shaft and crank pin ? _ 

Diameter of the shaft D = y'4o600X 3_ ^.g inches. 

4 


Diameter of the crank pin d — - — 7*63 inches. 

Zo 


Example 5. A steam engine of 3C8 horses is to make 32 revolutions per 
minute. Required the diameter of the main shaft? 

Diameter D — 5 inches. 

Example 6. A cog wheel of R = 6-5 feet radius is to gear with a pinion of 
r = 1 25 feet radius, and to transmit an effect of 231 horses with 42 revolutions 
per minute. Required the diameter of the wheel aud pinion shafts? The force 
E is acting uniformly at the periphery, 

3 ’ 231 
42 


Diameter of wheel shaft D — 4*35 
D:d = 


\7 


y r 


= 7*66 inches 


Diameter of pinion shaft d — 7*66 4 / ^ " ,r> ~ = 4*41 inches. 

V 6*5 







































Roofs of Wood and Iron. 


299 


ROOFS OF WOOD AND IRON. 



The Figs. 1 and 2 illustrate the common form of wooden roofs, as constructed 
over spans of from 30 to 80 feet. When the span exceeds 60 feet, a proportionate 
number of struts and tie-rods must be inserted, as shown by the dotted lines, or as 
illustrated for iron roofs. 


Table of Timber Dimensions, in Inches, for Roofs over Spans 

from 30 to 80 feet. 


Name of timbers. 

30 

35 

40 

45 

Span 

50 

in feet. 

55 

60 

70 

80 

Tie-beams, 

a 

5X6 

6X7 

6X8 

7X8 

8X9 

8X12 

9X11 

10X11 

10X12 

Truss rafter, 

b 

5X5 

5X6 

6X7 

7X7 

8X8 

8 X 9 

9X9 

9X10 

10X11 

Collar-beams 

c 

5X5 

5X6 

6X7 

7X7 

8X8 

8 X 9 

9 X 9 

9X10 

10X11 

Com. rafter, 

d 

2X5 

2X5 

2X6 

2X6 

2X6 

2 X 7 

2 X 7 

2iX 8 

3 X 9 

Purlins, 

e 

5X6 

5X6 

5X7 

6X7 

6X8 

6 X 8 

6 X 9 

6 X 9 

6 X 9 

Struts, . . 

f 

3X4 

3X5 

3X6 

4X7 

4X8 

5 X 8 

5 X 9 

6 X 9 

6 X 9 

King’s rod, 

h 

1 

1 

1 

n 

li 

If 

U 

11 

2 

Bolts, . . 


f 

f 

1 

f 

$ 

1 

u 

n 

If 


Lioad on Roofs in Pounds per Square Foot, exclusive of 

Framing. 


Lead covering. 

Pounds. 

. 8 

Tiles, .... 
Boarding, f thick, . 

Pounds. 

9 to 16 

Zinc covering, . 

2 

3 

Corrugated Iron,. 

. . 3.5 

Boarding, H thick, , 

6 

Slates, 

10 

Pressure of wind, . 

40 


In high latitudes the roofs may be covered with snow, which makes a pressure 
of 10 founds per square foot per foot of depth of the snow. 







































800 


Strain on Roofs. 



STRAIN ON ROOFS. 


The above figures illustrate four different kinds of pointed iron roofs, of which 
figure 3 is most in use. 

Fine Lines in Tension and Thick Lines in Compression. 

Notation of Letters. 


L — length of principal rafter. 

R = rise of roof above centre of tie- 
rod, which is generally one-fifth 
of the span. 

S =half the span, or one-half the 
length of tie-rod between sup¬ 
port on the walls. 

N= number of bays or divisions in the 
whole span, which is 8 in the 
diagram. 

s, s' and s" = compression on the cor¬ 
responding struts. 


W= load, uniformly distributed on 
the rafter, including weight of 
framing. 

C = compression at the end of prin¬ 
cipal rafter. 

T = tension at the end of tie-rod. 

c, cf and c"= compressions between 
the corresponding divisions of 
the rafter. 

t and V — tensions between the cor¬ 
responding divisions of the tie- 
rod. 


The formulas will answer for any units of weight and measure. 


Compression on 
Principal Rafter. 

Tension on Tie- 
rod. 

Tension on King and 
Queen Rods. 

Compression on 
Struts. 

LW 
' 2B ' 

c = C-<? 

ws 

2H * 

N 

s— 4W 

S N ' 

N 

C =°~N' 

1 

II 

Q^ W - 
* N 

, sw 

s 7 =-. 

N 

C " =r C— —. 

N 

II 

1 

rO 

II 

2.66 TF 

N ’ 


The principals to be placed not more than 7 feet apart. 





























Bridges. 


301 


BRIDGES. 



Fine Lines in Tension and Thick Lines in Compression. 


The Warren girder consists of fifteen equilateral triangles formed by the trusses 
and ties, which make eight divisions or bays in the span. 

The depth of the girder is 0.10825 of the span. 

The load uniformly distributed on each girder, multiplied by the tabular number, 
will be the strain on the corresponding part. 


Parts in Compression. 


Parts in Tension. 


Top-beam. 


s 

S l 

S* 

S 3 

t 


P 

0.5 

0.875 

1.125 

1.25 

0.8 

0.6 

0.4 

T 

T i 

rji-2 

T s 

s 

s 1 

s 2 


Ties. 


i 3 

0.2 

s 3 


Bottom Tie-rod. 


Trusses. 


Parts in Tension. 


Parts in Compression. 


Weight of" one pair of 'Warren’s Girders in tons, 

for a single track of railway on the top or on the bottom (approximate). 


On 





Span of the girder in feet. 




the 

50 

60 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

Top, 

11 

15 

18 

23 

27 

32 

38 

44 

51 

58 

66 

75 

Bottom, 

15 

19 

24 

29 

35 

41 

48 

56 

64 

72 

80 

89 


Talble of Dimensions in indies of Dolled Iron, 

for roofs on spans from 30 to 80 feet (figures 3 to 6, page 300). 


Name of 
iron. 

30 

s 

40 

pan in feet 

50 

60 

TO 

80 

Rafter T-iron, 

L 

2fX2fXf 

3fX3Xf 

4X-3fXf 

5X4fXf 

6fX5Xf 

6X6Xf 

Struts T-iron, 

s 

2fX2fXf 

3X21X* 

3X3X1 

4X4Xf 

4fX4fXt 

5X41X1 

King bolt, 

K 

1 

H 

If 

H 

If 

If 

Queen bolt, -j 

Q 

f 

T 

8 

1 

If 


If 

l 

O' 

t 

i 

¥ 

1 

If 

If 

Tie-rod, ( 

T 

H 

if 

if 

If 

If 

If 


f 

l 

if 

H 

H 

If 

If 

Weight, lbs., 


1500 

3000 

4800 

7000 

9550 

12400 


The last line shows the approximate weight in pounds of each principal when 
the rise of the roof is 0.2 of the span. 




























































502 


Suspension Bridges. 



SUSPENSION BRIDGES. 


Notation of Letters. 


IV= 
T = 
t = 

t' = 
S' = 
L = 

The 


total load on the bridge, 
tension on the chain in the centre, 
tension at any point of distance, 
Z>, from the centre, 
tension at abutments, 
span. 

length of chain between the 
piers. 


H= versed sine, or height of abutments 
above centre of chain. 

D and d = co-ordinates for any point of 
the chain. 

v = angle of suspension at abutments. 

(The angle of the counter-chain 
ought to be equal to that of sus¬ 
pension.) 


formulas will answer for any system of weights and measures. 

HD 2 • i 

T—^W cot.®. 


t 


=Wf) 


+ 1 . 


T= 


ws 




8H 
WS 
8 T' 
S 


H— —, generally. 

cotan.® =-- 

S 


L = K/S'H- i/G.25 S* + 9iP) 




w == 




sin.® 


V( 


4 #)+> 


ws rimy 
8H VI s ) 


+1- 


































Steam Hammeb. 


803 



BOLLMAN’S AMERICAN TRUSS BRIDGES. 

Notation of Letters. 


W = total load uniformly distributed on 
the bridge. 

w — load on each point of suspension. 

S == span. 

D and d — distances from abutments A 
and B to point of suspension. 

A and a — cross areas of the tension 
and counter-tension rods in 
square inches. 


R and r = lengths of tension and coun¬ 
ter-tension rods. 

H — depth of truss, which is usually 
one-seventh of the span. 

N = jiumber of points of suspension. 

T and t tensions on the rods R and r 
„ respectively. 

G = compression on the top at centre. 


These formulas will answer for any system of weights and measures. 

W m wDR wdr S W 

w = - 1 -- t -- (J =- 

N SH SH SH 


■ > 

When T and t are tons, A and a = square inches, D, d, S, H, R and r = feet, 
then 

. WDR Wdr ttt 5 aNSH lir 5 ANSH 

A = WSH- “ 6NJ3B W= ~d~r W =^NT\ 

\ STEAM HAMMER. 


A heavy steam hammer with short fall produces a better forging than a light 
hammer with a high fall, although the dynamical momentum may be the same in 
both cases. This is accounted for by the inertia of the ingot forged. 

The effect of blows of a heavy hammer and short fall will penetrate through the 
metal, and nearly with the same effect on the anvil side, while a light hammer and 
high fall will affect the metal on or near the surface of the blow, because most of 
the momentum is in the latter case discharged in the inertia of the iiigotforged. 
In forging a large shaft, it is generally piled up with iron bars sometimes rolled into 
a segment form to suit the pile. When placed under the hammer in a welding heat, 
very light and gentle blows are first given, then the momentum of a light hammer 
may be discharged in the bars nearest to the blow, while a heavy hammer will 
squeeze the whole mass together throughout, and a sound welding will be produced. 

The additional expense of a heavy hammer is fully compensated by the waste of 
labor and materials under a small one. I have often seen, in broken shafts, the 
bars in the centre as clear and unwelded as when first piled, which is a sure indi¬ 
cation that the shaft has been forged by a too light hammer. In crank-shafts for 
propeller engines, forged under a light hammer, when brought to the machine- 
shop the best part of the metal is worked away by planing and turning, and the 
poorest left for the engine; but if forged under a heavy hammer, the difference in 
quality of metal will not be so great. 

Cases of this kind are well known in the United States navy. 


Weight of Steam Hammers. 

The weight of a steam hammer in poiinds should be at least eighty times the 
square of the diameter of the shaft in inches. 



























304 


Gravitation. 


GRAVITATION. 

Gravity or Gravil ation is a mutual faculty which all bodies in nature 
possess, to attract one another; or Gravity is the force by which all bodies 
tend to approach each other. A large body attracting a comparatively very 
small one, and their distance apart being inconsiderable, the force of gravity in 
the small body will be very sensible compared with that in the large one ; such 
is the case with the body, our earth, attracting small bodies on or near her sur¬ 
face. 

Gravitation is not periodical, it acts continually ever and ever. A body placed 
unsupported at a distance from the earth, the force of gravity is instantly oper¬ 
ating to draw it down, and then we say, “ the body fell down ” If it were possi- 
i ble to withdraw the attraction between the body and the earth, it would not 
fall down, but remain unsupported in the space where it was placed;—giving 
; the body a motion upwards it would continue that, and never come back to the 
earth again. 

Law of Gravity. 

The force of Gravity is proportional to the mass of the attracting bodies, and in¬ 
verse as the square of their distance apart. 

This law was discovered by Sir Isaac Newton. It is this law that supports the 
condition of the whole universe, aud enables us to calculate the distances, mo¬ 
tions and masses, &c., of the heavenly bodies. 

The unit or measure of force of gravity is assumed to be the velocity a falling 
body has attained at the end of the first second it falls; this unit is commonly 
denoted by the letter g; its value at the level of the sea in New York is 
g = 32166 feet per second, in vacuum. The space fallen through in the first 
second is ?g = 16-083 feet. 

This value augments with the latitude, and abates with the elevation above 
the level of the sea. 

I — latitude, h = height in feet above the level of the sea, and r = radius of 
the earth in feet, at the given latitude l. 

r = 20887510(1+0-00164 cos.2 T), 

g = 32-16954(1 — 0-00284 cos.2Z)(l — —.) 

Letters denote. 

S — the space in feet, which the falling body passes through in the time T. 

u = the space in feet, which the body falls in the 2'th second. 

Y= velocity in feet per second, of the falling body at the end of the time T. 

T — time in seconds the body is falling. 

The accompanying Diagram is a good il¬ 
lustration of the acceleration of a falling 
body. The body is supposed to fall from a 
to b, every small triangle represents the 
space 16"08 feet which the body falls in 
the first second ; when the body has reached 
the line 3" seconds, it will be found that it 
has passed 9 triangles, and 9X16-08 = 144-72 
feet the space which a body will fall in 3" 
seconds. The number of triangles between 
each line is the space u which the body has 
fallen in that second. Between 3" and 4" 
are 7 triangles and 7X16‘08 = 112-56 feet, 
the space fallen through in the fourth sec¬ 
ond. Under the line 3" will be found 6 tri¬ 
angles, which represents the velocity V the 
body has obtained at the end of the third 
second or 6X16-08 = 96-48 feet per second. 
For every successive second the body will 
gain two triangles or 2X16-08 = 32-J6 feet 
per second. 



























Gravitation. 


805 


Formulas for Accelerated Motion. 

\ r 2gS = 8 . 021 /$, . 

V% 


V = g T = —■ 


s _gT* _ VT _ F 2 _ 
2 2 


T= V = 2 £ 
9 V 

g(T—k), 


u 


2 g 64.33’ 

2 S _ VS 
g 4.01’ * 

u 


1. 

2 . 

3. 

4 . 


r=-+*, . . . 

g 

Example 1. A body ig dropped at a height of 98 feet. What velocity will it 
have when it reaches the ground, and what time will it take to fall down ? 


Formula 1. 
Formula 3. 


V = 8.02j/& = 8.02) 55”= 79.39 leet per second, 

T = = 2.46 seconds. 

4.0 L 4.01 


Example. 2. A body was dropped at the opening of a hole in a rock, and reached 
the bottom after 3.5 seconds. How deep was the hole? 


Formula 2. 


2 


: 32 - i T > _ x . 8 - 52 = 196.98 feet. 
2 


Retarded Motion. 

A body thrown up vertically will obtain inversely the same motion as when it 
falls down, because it is the same force that acts upon it, and causes retarded 
motion when it ascends, and accelerated motion when it descends. 

V = the velocity at which the body starts to ascend. 
v = velocity at the end of the time t. 

T = time in seconds in which the body will ascend. 

I = any time less than T. 

S = height in feet to which the body will ascend. 
s = the space it ascends in the time t. 


Formulas for Retarded Motion. 

v = V- 


l 2 


g= vi-e^-t v + s£, 

A A 

V=v + gt= s +££, 

t A 


t = 


■v - V_ f 

g \ 


V s 

g* 


2s 


5. 

6 . 

7. 

8 . 


g g \ ff* g 

Formulas for T and S are the same as for accelerated motion. 

Example 3. A ball starts to ascend with a velocity of 135 feet per second. At 
what velocity will it strike an object 60 feet above? Find the time t by the 
Formula 8. _ 

,_J35_ Il35« __ 2 X 60 

32.16 \32.16 2 32 j ti — 0-41 seconds, 

until it strikes, and from Formula 5 we have 

v = 135 — 32.16 X 0.41 = 121.83 feet per second. , 


20 















306 


Gravitation. 


Example 4. A ball thrown up vertically from a cannon, occupied 20 seconds, 
nntil it arrived at the same place it started from. How high up was the ball, 
and at what velocity did it start ? 

One-half of 20 = 10 seconds. Formula 2. 


S-- 


32.16 x 10* 


= 1608 feet high. 


V — 32.16 x 10 = 321.6 feet per second. 

If a cannon-ball be shot from A, in the direction AB, at an angle BAO to the 
horizon, there are two forces acting on the ball at the same time, namely — the 
force of gunpowder, which would propel the ball uniformly in the direction AB, 
and the force of gravity, which only acts to draw the.ball down at an accelerated 
motion; these two different (uniform and accelerated) motions will cause the ball 
to move in a curved line (Parabola) AaC. Fig. 225. 
j V = velocity of the ball at A. W = weight of the ball in pounds. 

5 = the greatest height of ball over the horizontal line AC. 

t = time from A to C, via a. p — pounds of powder in the charge. 

6 = the distance from A to C, called horizontal range. 


V= 2800 


4 


p_ 

w’ 


WV 2 V 

p - -, 6 = 87.06 sln.x cos.x —. 

F 7840000’ W 


Example 5. The cannon being loaded sufficiently to give the ball a velocity of 
900 feet per second, the angle x = 45°. Required, the distance b =? and the 
time t = 'l 


6 = 


900' 2 x sin.45° x cos.45° 


= 1259 feet, the distance from A to C. 


32.16 

It will be observed that the distance b will be longest when the angle x is 45°, 
because the product of sine and cosine is greatest for that angle, sin. 45° X cos. 
45° = 0.5. 

Example 6. What time will it take for a ball to roll 38 feet on an inclined 
plane, angle x — 12°.20', and what velocity has it at 38 feet from the starting- 
point ? Fig. 222. 


T= 


4 ^- 4 -. 


2x38 


3.33 seconds. 


gsm.x \ 32.16 x sin. 12° 20 / 

V=g Tsin.x = 32.16 x 3.33 xsin.l2° 20' = 22.8 feet per second. 

Resistance of Air to the Flight of Projectiles. 

A = area of resistance of the projectile in square inches. 

<f> — angle of resistance of the projectile, which for flat surfaces sin 2 <£ = 1 , for 
sphere sin. 2 <£ = 0.5. 

For a pointed projectile of parabolic form, and when the ordinate is double the 
abscissa sin A<J> — 0.25.t 

V == velocity of the projectile in feet per second. 

R, = resistance to the projectile in pounds. 

72 =A V* sin. 2 r> 

57000 

Let T denote the time of flight in seconds, and W = weight in pounds of the 
projectile. 

D — distance in feet which the projectile is retarded by resistance of air in the 

time T. 

32.16672 T 2 1672 T 72 


7 ) = 


2W 


W 


A > 























































808 Table for Falling Bodies. 


V»lo- 

Space fall- 

[ 

Time in 

Velocity 

Space fall- 

Time in 

Velocity 

1 

Space fall. 

Time in 

at the 

•Ui. 

en through 

seconds. 

at the 
end. 

en through 

seconds. 

at the 
end. 

en through 

seconds. 

V 

S 

T 

V 

s 

T 

V 

s 

T 

o-l 

•00015 

0-0031 

5-1 

•40388 

0-158 

11 

1-8789 

0*342 

0.2 

•00062 

0-0062 

5-2 

•41987 

0-162 

12 

2-0652 

0-373 , 

0-3 

•00139 

0-0093 

5-3 

•43618 

0-165 

13 

2-6242 

0-405 

0-4 

•00248 

0-0124 

5-4 

•45279 

0-168 

14 

3-0435 

0-436 

0*5 

0*6 

•00388 

•00559 

0-0155 

0-0187 

5-5 

•46972 

0-171 

15 

16 

3-4938 

3-9751 

0-467 

0-498 

5-6 

•48695 

0-174 

0-7 

•00761 

0-0218 

5-7 

•50450 

0-177 

17 

4-4876 

0-530 

0-8 

•00994 

0-0230 

5-8 

•52236 

0181 

18 

5-0310 

0-560 

0-9 

•01257 

0-0280 

5-9 

•55057 

0-184 

19 

5-6056 

0-591 

1- 

•01552 

0-0311 

6- 

•55900 

0-187 

20 

6-2112 

0-622 

1-1 

•01879 

0-0342 

6-1 

•57779 

0-190 

21 

6-8478 

0-654 

1*2 

•02065 

0-0373 

6-2 

•59689 

0-193 

22 

7-5155 

0-685 

1-3 

•02621 

0-0404 

6-3 

•61630 

0-196 

23 

8-2143 

0-716 

1-4 

•03043 

0-0436 

6.4 

•63602 

0-199 

24 

8-9441 

0-747 

1-5 

•03493 

0-0467 

6-5 

•65606 

0-202 

25 

9-7049 

0-778 

1-6 

•03975 

0-05 

6-6 

•67639 

0-205 

26 

10-497 

0-810 

1-7 

•04487 

0-052 

6:7 

•69705 

0-209 

27 

11-320 

0-840 

1-8 

1-9 

•05031 

•05605 

0-556 

0-0591 

6.8 

6-9 

•71801 

•73928 

0-212 

0-215 

28 

12.174 

0-872 

0-903 

ZV 

13’059 

2- 

•06211 

0-0623 

7- 

•76087 

0-218 

30 

13-975 

0-933 

2-1 

•06S47 

0-0654 

7.1 

•78276 

0-221 

31 

14-922 

0-965 

2-2 

•07515 

0-0685 

7-2 

•80497 

0-224 

32 | 

15-900 

0-996 

2*3 

•08214 

0-0717 

7-3 

•82748 

0-227 

33 

16-910 

1-025 

2-4 

•08944 

0-0747 

7-4 

•85031 

0-231 

34 

18-789 

1-058 

2-5 

•09705 

0-0780 

7-5 

•87344 

0-234 

35 

19-022 

1-091 

2-6 

•10497 

0-0810 

7-6 

•89689 

0-237 

36 

20-124 

1-120 

2-7 

•11320 

0-0841 

7-7 

•92065 

0-240 

37 

21-258 

1-151 

2-8 

•12174 

0-0872 

7-8 

•94472 

0-243 

38 

22-422 

1-184 

2-9 

•13059 

0-0903 

7-9 

•96910 

0-246 

39 

23-618 

1-213 

3- 

•13975 

0-0934 

8- 

•99379 

0-250 

40 

24-844 

1-243 

3-1 

•14922 

0-0966 

8-1 

1-0187 

0-253 

41 

26-102 

1-276 

3-2 

•15900 

0-0997 

8-2 

1-0441 

0-256 

42 

27-391 

1-308 

3-3 

•16910 

0-1025 

8-3 

1-0697 

0-259 

43 

Oo.k't 

1*338 




3-4 

•18788 

0-1059 

8.4 

1-0956 

0-262 

44 

30-062 

1-370 

3-5 

•19022 

0-1092 

8 5 

1-1218 

0-265 

45 

31-444 

1-400 

3-6 

•20124 

0-1121 

8.6 

1-1484 

0-268 

46 

32-857 

1-431 

3*7 

•21257 

0-1152 

8-7 

1-1753 

0-271 

47 

34-301 

1-463 

3-8 

•22422 

0-1185 

8-8 

1-2015 

0-274 

48 

35-776 

1-495 

3-9 

•23618 

0-1214 

8-9 

1-2299 

0-278 

49 

37-282 

1-525 

4* 

•24844 

0-1246 

9- 

1-2577 

0-281 

50 

38-820 

1-555 

4*1 

•26102 

0-1278 

9-1 

1-2858 

0-283 

51 

40-388 

1-588 

4*3 

•27391 

0-1309 

9-2 

1-3143 

0-287 

52 

41-987 

1-619 

4*3 

•28571 

0-1339 

9-3 

1-3430 

0-290 

53 

43-618 

1-650 

4*4 

•30062 

0-1371 

9-4 

1-3720 

0-293 

54 

45-279 

1-680 

4*5 

•31444 

0-1403 

9-5 

1-4041 

0-296 

55 

46-972 

1-711 

4*6 

•32857 

0-1433 

9-6 

1-4310 

0-300 

56 

48-695 

1-742 

4*7 

•34301 

0-1465 

9-7 

1-4610 

0-302 

57 

50-450 

1-774 

4.8 

•35776 

0-1496 

9-8 

1-4913 

0-306 

58 | 

52-236 

1-805 

4*9 

•37282 

0-1526 

9-9 

1-5219 

0-309 

59 j 

55-058 1 

1-835 

&• 

•38820 

0-1559 

10 

1-5528 

0-312 

60 1 

55-900 

1-868 



























Dynamics of Mass and Weight in Moving Bodies. 


309 


DYNAMICS OF MASS AND WEIGHT IN 
MOVING BODIES. 



Let a constant force, F, be applied to a body, W, free to move, then the body will 
start and continue with an accelerated velocity until the force ceases to act, when 
it will continue in the same direction with a uniform velocity equal to that of the 
final action of the force. Any force, however small, is able to set in motion any 
body free to move, or to bring to rest any moving body, however large. 

The direction of the force F must pass through the centre of gravity of W, 
otherwise the body will be set in rotation. 

A force applied obliquely from space to the surface of the earth would change 
the axis of rotation, which is actually the case with meteors falling on the earth. 

No force is required to maintain a uniform motion of a body free to move; but 
force is required to bring a body from rest into a uniform motion. If force i3 
applied to maintain a body not free to move in uniform motion, such force is 
expended in overcoming the friction and resistance of the medium in which the 
body moves. A steamboat or a railway train in motion is thus suspended between 
the action of two opposite forces—namely, the driving force on the one side, and 
the friction and resistance on the other. When the opposite forces are equal, the 
motion will be uniform; and any change of velocity is due to a disparity between 
these opposing forces. 

Mass means the real quantity of matter in a body. It is proportionate to 
weight when compared in one locality. 

The mass of a body is a constant quantity, whilst the weight varies with the 
foi’ce of attraction or gravity. A body weighed on a spring balance at the equator 
will weigh less than if weighed at the poles, because the radius of the earth is 
greater, and consequently the force of gravity is less at the equator. 

A body weighing one hundred pounds at the surface of the earth would weigh 
only twenty-five pounds at a height of 3956 miles (radius of the earth) above the 
surface, whilst the mass of the body remains constant. 

The weight of a body is inverse as the square of its distance from the centre of 
the earth when weighed above the surface; but if weighed below the surface, the 
weight will be only 

W r 



W — weight at the surface; w — weight below the surface; R = radius of the 
earth, and r = radius where the body is weighed. 

A body weighed at a height h above the surface of the earth would weigh 

WE 2 

w =-, 

(-B + A) 2 

and the force of gravity, or acceleratrix g, at the height A, is 

_ 32.166.Z? 2 
9 ~ (i2 + A) 2 ' 

Therefore the mass M of a body is 

w 

M=~, which is a constant quantity. 

9 

Below the surface of the earth the acceleratrix g is 

32.166r 
R 


9 = 









810 


Dynamical TFrms. 


Attraction of tlie Sun and Moon on tlxe Eartlx, 

Assuming the mean radius of the earth as a unit for the distances to the sun and 


to the moon, and the attraction in pounds per 100,000 pounds of material 
earth, then the data will be as follows: 

in the 

Location of Attraction in the centre or ends of 

Attraction of the 

Distances to tli& 

the diameter of the earth pointing to the sun 

Sun. 


Sun. 


or moon. 

Moon. 

Moon. 

In the centre of the earth, or where the sun 1 

11.3273 

0.0769971 

24000 

60 

or moon is seen to set or rise, . . . j 




On the surface of the earth nearest to the) 

11.3283 

0.0794466 

23999 

59 

sun or moon, .. j 





On the surface of the earth farthest from) 

11.3264 

0.0744936 

24001 

61 

the suii or moon,.j 





Greatest difference, . . . 

0.00190 

0.0049530 

2 

2 


It is these differences of attraction which, in co-operation with similar differ¬ 
ences of centrifugal forces, cause the tidal waves of the ocean. Although the sun’s 
attraction on the earth is one hundred and ninety-five times that of the moon, its 
difference of attraction in the diameter of the earth is only one-half that of the 
moon, for which reason the moon causes the highest tidal waves. 

A weight of 100,000 pounds weighed on a spring balance at sunrise would weigh 
only 100,000 —11.328 = 99,988.67 pounds where the sun is in the zenith; and if 
the same weight is weighed at midnight in a latitude opposite the sun, it would 
weigh 100,011.328 pounds on the same spring balance. But when substances are 
weighed by weights, there will be no difference where they are weighed. 

Dynaixxic Momentum in a body free to move is of two kinds—namely, 
Momentum of time F T = M V momentum of motion. 

A force multiplied by its time of action on a body free to move is the momentum 
of time , which is equal to the mass of the body multiplied by its uniform velocity 
after the force has ceased to act, the momentum of motion. 

Vis-Viva, or living force , is a term intended to express the quantum of work 
concentrated in a moving mass, and is usually denoted by M V s , which is twice the 
true amount of work; but as there is no living force in a dead body, the term is 
improper and confusing. See Journal of the Franklin Institute for i864 and 1865. 

Moment of Inertia is another confused term not always properly under¬ 
stood. In substance it means work imparted to or given out by a revolving body. 
It is denoted by M iE 2 , in which M — the mass, and R = radius of gyration. 

Inertia in a body free to move is equal to the force applied to change its 
motion. 

A force multiplied by the lever it acts upon is called static momentum , which is 
analogous to the force of inertia multiplied by the radius of gyration in a revolv¬ 
ing body, which seems to have a better claim to be called moment of inertia. 

A List of Confused Dynamical Terms, 

which ought to be abolished in our school-books. 


Quantity of motion. 

Total quantity of work. 
Actual total amount of work. 
Virtual velocity. 

Heat a mode of motion. 


Rate of work (which is power). 
Quantity of moving force in a body (in¬ 
ertia or work). 

Vis-viva and principle of vis-viva. 
Moment of inertia (if you please). 


See Nystrom’s Elements of Mechanics, which establishes strict precision in 
the meaning of dynamical terms, and abolishes the ideal vocabulary heretofore 
used in text-books on Mechanics. The subject of confusion in dynamical terms 
has been thoroughly discussed, and may now be considered exhausted. 





















Dynamical Terms. 


311 


Proper Dynamii al Terms. 

All the terms necessary in statics and dynamics are as follows: 

Elements. Functions. 

Force = F. Power, P == F V. 

Velocity = V. Space, S = VT. 

Time = T. Work, W = F V T. j 

Static momentum = FI. 

Dynamic momentum, F T = M V. 

These terms include all cases in dynamics, whether by mechanical force, gravity, 
uniform accelerated, retarded, straight or curved linear motion, k. 

In the static momentum FI, the lever l may mean radius of gyration in revolv¬ 
ing bodies. 

In irregular motion, V means the mean velocity in the line T. 

Force is any action which can be expressed simply by weight, such as pressure, 
attraction, repulsion, gravity, inertia, exertion, cohesion, electricity, magnetism, strain, 
stress, strength, thrust, burden, load, squeeze, pull , push, resistance, compression, etc. 

Power is any action of force and velocity , such as dynamic momentum of 
inertia, impetus, dynamic effect, traction, propulsion, impulsion, labor, etc. 

Worlt is any action which includes the three simple elements force, velocity 
and time, such as energy, vis-viva s, labor, throw, cast , haul, drag, draw, occupation, 
exercise, idling, lift, raise, heave, cultivate., to tilt, etc. 

The term effect is used in three senses—namely, effect of force, effect of power and 
effect of work. 

'■ Much inconvenience has been experienced for the want of distinction between 
elements and functions in dynamics, and sufficient room cannot be allowed for a 
full explanation of the subject in this Pocket Book. In the Elements of Mechan¬ 
ics before mentioned the subject is fully elucidated. 


The 'Worlt concentrated in a moving body is equal to the work expended 
in bestowing its motion, and is equal to the work required in bringing the body to 
rest, which is derived from the primitive formula F VT; but for accelerated motion 
V means the mean velocity in the time T, which is just one-half of the final 
velocity V, when the acceleratrix G or g is constant. 

The following table of formulas will show what a variety of problems are con¬ 
nected with a force acting on a body free to move. 

: When a body is left free to the action, of gravity in killing or rising, the accel¬ 
eratrix G = g, and the force F— W. 

; Example 1. What force F=l is required to give a body IF = 1689 pounds a 
velocity of V— 36 feet per second in a time T = 5.6 seconds? 

Find in the formulas under constant force the one which contains the given 
quantities W, V and T, which is the second formula. 

^ w V 1689 X 36 00 „ ctr , 4l 

F — -=-= 387.55 pounds, the answer. 

g T 32.166 X 5.6 r , 

Example 2. A projectile of W = 150 pounds is fired horizontally from a rifled 
gun of S= 11 feet in length, in which it receives a velocity of V = 950 feet per 
second. Required, the mean force F = ? of the powder acting on the projectile, 
when the friction in the rifle is 230 pounds. 

W V 2 150 V 950 2 

= — ^ — = 191302-1-230 = 191532 pounds, 

2 gS 2X 32.166X 11 

the force required. 








312 


Moving Bodies. 


Example 3.—The moving parts in a propeller steam-engine, such as the steam- 
piston, piston-rods, cross-heads, connecting-rod, &c. &c., weigh W = 8456 
pounds. Stroke of piston = 4 feet, making n = 52 revolutions per minute. 
What force F is required for each stroke, to set in motion and bring to rest the 
moving mass ? 

The velocity of the moving mass at half stroke will be (formula , page 263) 


2nrn _ 2 X 
60 


3-1416 X 2 X 52 
60 


= 10-79 feet per second. 


The time for each half stroke will be 


T = 


60 


= 0-28846 seconds. 


4 X 52 

Then the required mean force of the momentum will be 
WV 8456 X 10-79 
~g~T ~ 32-166 X 0-2846 


F= 


= 9966-8 pounds. 


For high grade of expansion of steam, this force acts beneficially to the move¬ 
ment of the engine. 

Example 4.—The mean force of gunpowder in a rifled gun is known to be 
231400 pounds, on a projectile W = 180 lbs. The friction of the projectile 
through the gun is estimated to 264 pounds, leaving F — 231400 — 264 = 231136 
pounds. The length of the gun is S = 12 feet, elevated to an angle x = 6° 30'. 
Required the velocity V= ? of the projectile when it leaves the gun. 



I 2 X 32-166 X 12 (^T ~ 8in - 6 ° 3 °') 


= 995-64 feet per second, the answer. 

Example 5.—What velocity V— ? can a steam-engine of H = 56 horses im¬ 
part to a body W — 9 tons in a time T— 30 seconds ? 


P = 56 X 550 = 19800 effects, and W— 9 X 2240 = 20160 lbs. 




2 X 32-166 X 19800 X 30 
20160 


= 43-538 feet per second. 


Example 6.—A body W — 3685 lbs. is moving with a velocity V — 56 feet per 
second. What time T = ? is required to bring that body to rest, with a force 
F — 128 pounds ? 



3685 X 56 
32166 X 128 


= 50-121 pounds, the answer. 


Example 7.—What power P = ? is required to drive a centrifugal gun to 
throw out balls of W= 50 lbs. every T = 8 seconds, with a velocity V = 785 
feet per second (friction omitted) ? 


P = 


W V s 


50 X 7852 


2 g T 2 X 32-166 X 8 
divided by 550 = 108-85 horses, the power required. 


= 59867 effects, 


Example 8.—A sledge of TF== 20 lbs. strikes a spike into a log S = 0-08 foot, 
with a velocity of F= 25 feet per second. Required the force F = ? with which 
the spike was driven into the log, omitting the weight of the spike. 


„ W V 2 

F— - = 

2gs 


20 X 252 

2 X 32-166 X 0-08 


2628-9 pounds. 


Example 9.—A body starts to ascend vertically with a velocity of 860 feet per 
second. What will be its velocity at the end of T = 5 seconds ? 

V — G T = 32-166 X 5 = 160-830 feet per second, 
and 860 — 160-83 = 699-17 feet per second, the answer. 




















Dynamical Formulas. 


313 


Dynamical Formulas for Accelerated or Retarded 

Motion. 

Constant Force in Pounds acting on a Body free to move. 

2 PW 2 K K 
g T ~ G T 2 ~ S' 


F= 


GW 


wr 

g T : 


2 WS 
9 T a 


W V 2 
2 gS 


F T 
S 


\ 


Final Velocity in the Time T, or Uniform Velocity of a Moving Body. 

P T 



=-,/2 <?£=—= 


\2gPT_ frgK 

W \ W 


Time in Seconds in which the Force acts on the Body free to move. 


V _ WV 12 WS I2S _ 2FS K _ 2 S W I2WK 
G~gF~^jgF ~ ^ G ~ V K ~ P ~ g TK~yJiF* 

Constant Acceleration of the Force Fin Feet per Second, 
g F 2 S V V 2 gPT_FV*_gK 21 
G ~ W ~ T 2 ~ T ~ 2 S ~~ WS “ PT ~ WS~ FT*' 

Space in Feet in which the Force acts on the Body free to move. 

G T 2 V T V 2 gFT 2 PT gPT 2 g K K 

~ 2 2~ ~ 2 G~ 2 W ~ F ~ W V ~ GW ~ ~F~' 


Weight in Pounds of the Moving Body. 
gF gFT 2 2 gFS gFT gPT 3 g F 2 T 2 gK g T 2 K 

W ~ G ~ 2 S F2 — F “ 2 2 i J V* ~ 2 S 

Mean Power in Effects during the Time T, or in the Space S. 

FS g F 2 T 2 WS 2 W V 2 2K TK VK F V 2 
1 f 2 >F — g T 2 ~ 2g T — T ~ 2 S ~ 2 ~~ G T 

Work in Footpounds concentrated in a Moving Body. 


K= F S = 


W V 2 FVT G WVT F G T 2 g F 2 T 2 2 S P 


2 W 


2 g 2 2 g 2 

The Body moving in an Inclined Direction of an 

Angle x. 

Applied Constant Force in Pounds. 

F=w(-^±sm*) = w(^±sin.x) =» W ( T ^±sin.x). 


PT. 


Final Velocity in Feet per Seconds when the Force F ceases to act. 


(F . \ 2 S sin.a; I 77 \ 

V =9 T T sin.*; = —yr— = -J 2 g S (- T sin.x ). 

• \ 

Time of Action in Seconds. Acceleration. 

WV _ I 2 WS , iff _ . 

g^F^f TFsin.x) -xj g^Fr IF sin.x) & Wf + sin - a; 

Space in Feet. Work done by F. 


Bin - X )• K = WS ^ Bin.x) = sin.x). 

Use the upper sign when the direction of motion rises above the horizon, and 
the lower sign when the direction of motion dips under the horizon. 










































314 


Fly-Wheels. 


Force and Work in Revolving Bodies. Centre of 
Gyration. Fly-'Wlieels. 

Centre, of gyration is a point in revolving bodies in which, if all the revolving 
matter were there collected, it would obtain equal angular velocity from, and 
sustain equal resistance to, the force that gives it the rotary motion. 

The centre of gyration in different forms of bodies will be found by the for¬ 
mulas on pages 204 and 205. 

F== constant force in pounds, acting to l-otate the body as in figs. 249 and 
250, or the mean force on a steam-piston, 
r = radius in feet upon which the force F acts. For a steam-engine the mean 
radius will he r = 0-63661 X the radius of the crank, or 0.3183 S, when 
<S = stroke of* the steam-piston in feet. 

W = weight in pounds of a fly-wheel, or other rotating body. 
x — radius of centre gyration in feet. 

T — time in seconds in which the force F’is applied from the first start, or 
the time in which the velocity is accelerated. 

N = number of revolutions in the time T. 
n = number of revolutions per minute. 

K = work concentrated in the revolving body. 
f = irregularity in a fraction of the mean revolutions n. 

For a double-acting single-cylinder engine, the fly-wheel in its regular course 
of running has an irregular velocity through each revolution. Its smallest 
velocity is when the crank is at an angle of 40° from the beginning of the 
stroke, and its greatest velocity when at 40° from the end of the stroke. The 
larger the fly-wheel is for a given velocity, the more regular will the machinery 
run without limit. But the fly-wheel may be made so small that its accumu¬ 
lated work cannot carry the machinery around, which will be the case when, 
the irregularity /= 1. In ordinary practice make irregularity f — 0-1 to 0-01. 

Example 1.—What force F = ? is required to give a body W = 3600 pounds 
an angular velocity n = 76 revolutions per minute in a time T= 24 seconds, 
the radius of gyration being x — 12 feet, and the force F acting on a radius 
r =s 3 feet? Wx*n 3600 X 122X76 c 

F== 307 TT? = 307^1 X 24 X 3" = 1779 ' 5 P ° UndS > the answer * 
Example 2.—Required the weight W = ? of a fly-wheel for an engine of 
D — 36 inches diameter of cylinder double acting, with steam-pressure p — 50 
lbs. per sq. in. S = 6 feet the stroke of piston. Area of steam-piston 1017-8 
sq. in., and the force F = 1017-8 X 50 = 50890 pounds. Radius of gyration 
x = 10 feet, and n = 48 revolutions per minute. Assume f— 0-05. 


W — 


2542 FS 2542 X 50890 X 6 


= 67376-2 pounds, the weight required. 


x a f 482 x 102 x 0-05 

Should the steam be used expansively, the fly-wheel ought to be so much 
heavier, as the initial pressure is greater than the mean pressure. 

The radius of gyration in a fly-wheel, including the arms, can in practice be 
assumed to be the inner radius of the ring. 

Example 3.—What time from the start of engine is required to give the fly¬ 
wheel in Ex. 2 a velocity of n= 48 turns per minute? r — 0-3183 S= 1-9098 ft. 
m Wx 2 n 67376-2 X 102 X 48 _ 0 / 

307-17 F r 307-17 X 50890 X 1*0098 1085 seconds. 

Example 4.—Let the steam-engine in the preceding examples be applied to a 
rolling-mill, geared two to one of the rollers. An iron plate is rolled through with 
N — 8 revolutions of the engines, after which the revolutions were found to be 
reduced to % = 36 per minute. Required the work done in rolling the plate; 
and what time is required for the engine to regain the n = 48 revolutions? 

Work done by engine, K—2FSN— 2 X 67376-2 X 6 X 8 = 6468115-2 footps. 

Work done by fly-wheel, 

W x * ( W 2 _ „ S) 67376-2 X 102 (482 — 362) 

-- -— llo7671 footpounds, 


K = 


5866-5 


5866-5 


to which add 6468115-2 = 7625786-2 footpounds, work consumed in rolling plate. 
./The time required for the engine to make up the n = 48 revolutions will be 

T _ W x% (n — n x ) _ 67376-2 X 10 2 (48 — 36) 

~ 307-17 Fr ~ 307-17 X 50890 X 1-9098 


= 2-71 seconds. 














Circular Motion. 


315 


Formulas for Accelerated. Circular Motion* 

Force F, in pounds, acting on the. Lever or Radius r, to rotate the Body. 
W x 2 n W x* N 60 K K 

“ 307-17 Tr ~ 2-56 T 2 r~ ir r n T ~ 2 n r N 

Final Revolutions per Minute in the Time T. 


120 N 307 


_ Tn _ 2-56 FT 2 r 
N ~ 120 ~ Wx 2 


•17 F T r 60 K 1 5866-5 K 

Wa? ir r T F •\j IFx 2 

Total Number of Revolutions in the Time T. 

K T IT 

~ 2 n r F~ 1-566 art/ W‘ 


Time of Acceleration, in Seconds, from the Start of Change of Motion. 

wx 2 n _ 60 k _ x ywT . 


W x e n _ I 
T ~ 307-17 F r~ -l/” 


2-56 Fr tr r n F 2 Fr 
Radius of Gyration, in Feet, of the Revolving Body. 


V 307-17 FrT _ I 2- 

wn 7-\L 


I 

2-56 Fr T 2 

KT 

327-78 K 

J 

WN 

4N^WNFr n 

yWnT Fr 


W— 


Weight, in Pounds, of the Revolving Body. 

307-17 T Fr 2-56 T 2 Fr 5866-5 K K T 2 


x 2 n x 2 N n 2 x 2 2-454 x 2 N 2 

Work in Footpounds, concentrated in a Revolving Body. 
2-454 W x 2 N 2 ir r n F T 


_ W x 2 n 2 
~ 5866-5 


T 2 


60 


= 2nr NF 


Fly-Wheels for Steam-Engines. 

Fly-Wheel for a Single Acting Steam-Engine for Uniform Work. 


W = 


5866-5 F S 76-6 IPS 76-6 FS 5866-5 F S 

~ vfix 2 f x ■\j Wf ’ X « A/lF/' / x 2 rfiw’ 

' 

Fly-Wheel far a Double Acting Steam-Engine for Uniform Work. 

_ 50-42 [FS ^ _ 25i2 FS 
V ~~ n yjwf’ f ~ x 2 n 2 W * 


m 2542 FS 50.42 

~ rfl x 2 f ’ n x 


Fly-Wheel for a Double Acting Two-Cylinder Engine for Uniform Work. 

1172 FS 


_ 1172 FS _ 34-23 / FS _ 34-23 I FS 
~ rflaflf n ~ x ^ Wf X ~ n Wf *'~ 


x 2 n 2 W 











































816 


Centre of Gyration. 















































Centre of Gyration. 317 


af 

. 

245. Cone. 

/2A*+3 R' 

V 20 * 

T _ , /12/t 2 +3i2 2 

V 20 * 


v 

246. Conic Frustum. 

#=\ / h (R*i-3R r+R r*\ , 

V 10 1 R'+Rr+r* ' + 

klz r r .) 


« * 

l if .1 

247. 

Cylinder and Sphere. 

x = V a*+$r*, 
s= V a 5 +|r a . 

£ 

fE=3- 

248. 

Wedg-e awe? Ring. 

' O 

x = 0*204 N /l2/»+5 a +5% 

* = \/-2-- 



249. 

*Yy FAcc/. 

‘-s/W- 

FG: Wg = x*:s\ 



<= r\i ® Yr ) >) 1 

[i? \ 

250. Fly Wheel with Arms. 

R'+r* 4 r'+b* 

x\ W+w ) = W —g— +«o 12 

/6TT(K a +r a )+M?(4r’+i*) 

* V 12 (W+w) 














































318 

r' - 


Centrifugal Force. 


CENTRIFUGAL FORCE. 

Central Forces are of two kinds, centrifugal and centripetal. 

Centrifugal Force is the tendency which a revolving body has to 
depart from its centre of motion. 

Centripetal Force is that by which a revolving body is attracted or at¬ 
tached to its centre of motion. 

The Centrifuged and Centripetal ferrets are opposites to each other, and when 
equal the body revolves in a circle; but when they differ the body will revolve 
in other curved lines, as the Ellipse, the Parabola, &c., according to the nature 
of the difference in the forces. If the centrifugal force is o while the other is 
acting, the body will move straight to the centre of motion; and if the centripe¬ 
tal force is o while the other is acting, the body will depart from the circle in a 
straight line, tangent to the circle in the point where the centripetal force ceased 
to act. The central forces are distinct from the force that has set the body in 
motion. 

If the centrifugal force be made use of to produce an effect, such effect will be 
a>t the expense of the one producing the rotary motion. 

Letters denote. 

F — Centrifugal force in pounds. 

M = the Mass or weight of the revolving body in pounds. 

v = Velocity of the revolving body, in feet per second. 

R — Radii of the circle in which the body revolves, in feet. 

n = number of revolutions per minute. 

Example 1. Required the centrifugal force of a body weighing 63 pounds, and 
making 163 revolutions per minute, in a circle of 4 feet, 4 inches radius ? 

M Rn* 63X4-33X1633 


2933 


2933 


— 2475 pounds. 


Example 2. 
115 feet radii. 


A Railroad train runs 43 miles per hour on a curved track of 
What should be the obliquity of the track ? 


, Miles® 

tan.* = 

69i2 


43® 

69X115 = °’ 233 ’ 


or x = 13° 10', the obliquity of the track. 

Example, 3. A governor having its arms l = 1 foot, 6inches, how many revol¬ 
utions must it make per minute to form an angle x — 30° ? 

54-16 

n = —■ — =- = 47’5 revolutions per minute. 


j/l-5Xcos.30° 


: 227- 


M w 5 


M v" 1 





g R 32-16-R* 
4 M R it 1 n Q 


60 *- 


M Rn* 
2933~ 


Fg R 2933 F 
M = —— —- 


R 


n 


* M ? 
Mv 2933 F 


F g M n 

O 


2933 F 

WIT’ 






FRg 
M ’ 


1, 

2 , 

3 , 

4 , 

5 , 



























Centrifugal Force Governors. 


315) 



228. 


Centrifugal foixe of a ring. 


F = 


M n* V’/fr+r* 
4150 



229. 

Centrifugal force of a grinding stone , 
circle-plane , cylinder , rotating round 
its centre. 

p _ M Rn* 

~ '4150"* 



230. 

Centrifugal force of a cylinder rotating 
round the diameter of its base. 


F= 


Mn 2 l 
5866' 



231. 


Centrifugal force of a ball, 
„ Mn* R 
F = '293T- 



232. 


Governor. 

60 f9 54-16 54-16 

2 n 


di 


Vh Vlcosx* 


h = 2933 l = _ 2933 __ = h 
~~ n* ’ n 3 cos.# cos.#’ 


COS. a; = = y, r = \// a — A*. 

w n 3 t / 


































320 


Pendulum. 


PENDULUM. 


Simple Pendurnm is a material point under the action of gravitation, 
and suspended at a lixed point by a line of no weight. 

Compound Pendulum is a suspended rod and body of sensible mag¬ 
nitude, hxed as the simple pendulum. 

Centre of Oscillation is a point in which if all the matter in the com¬ 
pound pendulum were there collected, it would make a simple pendulum oscil¬ 
late at the same times. 

Angle of Oscillation is the space a pendulum describes when in mo¬ 
tion. 

The velocity of an oscillating body through the vertical position, is equal to 
the velocity a body would obtain by falling vertically the distance versed sitie of 
half the angle of oscillation. 

Letters denote. 

I = length of the simple pendulum, or the distance between the centre of sus¬ 
pension, and centre of oscillation in inches. 

t = time in seconds for n oscillations. 

n = number of single oscillations in the' time t. 

Example!. Required the length of a pendulum that will vibrate seconds? 
here n — 1, and t = 1". 

£2 

l — 39-109 — = 39-109 inches, the length of a pendulum for seconds. 

Example 2. Require the length of a pendulum that will make 180 vibrations 
p*»r minute ? here t — 60" and n = 180. 


I = 


39-109*® 39-109X60® . . , 

— — = -= 4*346 inches. 

tj® 180® 


Example 3. 
in 8 seconds ? 


How many vibrations will a pendulum of 25 inches length make 


6-254* 




VI 


6-254X8 x . 

— = 10 vibrations. 


]/25 


Example 4. A pendulum is 137‘C7 inches long and makes 8 vibrations in 15 
seconds. Required the unit or accelleratrix g = ? 


9 = 


0-8225 1 7i® 0-8225X137-67 X8* 


*® 


15® 


= 32-209. 


Example 5. A compound pendulum of two iron balls P and Q, having the 
centre of suspension between themselves: see Fig. 238. P = 38 pounds, Q = 12 
pounds, a = 25 inches, and b = 18 inches. How long is the simple pendulum, 
and how many vibrations will the pendulum make in 10 seconds ? 


a P —b Q 25X38 — 18X12 


1 = 


P+Q 

a® P+5® Q 


38+12 
25®X3S+18«X12 


= 14-68 inches. 


x(P+ Q) 14-68(38+12) 

the length of the single pendulum. 


= 37-68 inches, 


n — 


6-2541 6-254X10 

yf~ ** i/37-68" 


= 10-193 vibrations in 10 seconds. 


If a compound pendulum is hung up at its centre of oscillation, the former 
centre of suspension will be the centre of oscillation, and the pendulum will 
oscillate the same time. 


















Pendulum and Centre of Oscillation 


321 






233. 

Simple Pendulum. 


i !2 gj_ 

1 a 


39-1;* 

a 


7t a n‘ n 

ns/1 


t = 


6.25’ 

6 254; 

ST' 


-*-?r 1 

A A - 

l 

% 

A 

_y 

B 



0 


234. 

A = centre of grav 
it y. 

B = centre of gyra¬ 
tion. 

C centre of oscil? 
lation. 

a : 5 = b : l, 
b= s/aT^ 1-1432a, 
l = 1 ha. 


a/ 



236. 

I n* n a 

8 = 12; a ’ 

0-8225/n a 
8 ’ 

o — centre of suspen¬ 
sion, 

i 2r 

5a 



237. 


; _ a a P+5* Q 

TF+TQ' 

P and Q expressed 
in pounds, or cubic 
contents. 



235. 

Compound Pendu¬ 
lum. 

r = radius of cylin¬ 
der. 

16a*-i-3r a 

12a ’ 

, 4a r a 

1 - 3 + r a - 



238. 


aP — bQ 
*“ P+Q • 

} _ a*P+5 a Q 
" if(pVQT- 


Length of a Pendulum vibrating second* at the level of the sea, in various places. 
At the Equator, lat. 0° O' 0" - - . • * - 39-0152 inches. 

“ Washington, lat. 38° 53' 23" ..... 39-0958 “ 

“ New York, lat. 40° 42' 40" - * 39-1017 “ 

“ London, lat. 51° 31' ----*** 39-1393 “ 

« lat. 45°. 39-1270 “ 

« Stockholm, lat. 59° 21' 30". 39-1846 “ 

l = 39-127 — 0*09982 cos.2 lat. for seconds. 


21 






















































822 


Collision of Bodies in Motion. 


COLLISION OF BODIES IN MOTION. 

When bodies in motion come in collision with each other, the sum of 
their concentrated momentum will be the same after the collision as 
i before, but their velocities and sometimes their directions will differ. 

On the accompanying page the bodies are supposed to move in the 
same straight line, and the formula illustrates the consequences after 
collision. 

Letters denote. 

M and m = weight of the bodies in pounds. 

F and v = their respective velocities in feet per second. 

V and 1 / = respective velocties of the bodies after impact. 

K and k = coefficient of elasticity, which for perfectly hard bodies k—0 
and for perfect elastic bodies &=1, therefore the elastic coefficient will 
always be between 0 and 1. When the bodies are perfectly hard their 
velocities after impact will be common. 


For M, K = 


MV 


M(F— F')’ 


For in, k = - 


mv 


m (v— V') 


Example 1. Fig 191. The non-elastic body weighs M =25 pounds, and 
moves at a velocity F=l2 feet per second; to= 16 pounds, and v—9. Re¬ 
quired the bodies’ common velocities, v'—1 after impact. 


V '_ MV-}~mv 


M-j-m 


26X12-|-16X9 _ 10>83 f ee j. r gecond. 
25+16 


Example 2. Fig. 196. The perfect elastic body Af=84 pounds, F=18, 
w--=48, and u=27. Required the velocity V'=1 after impact with the 
body m. 


F= 


18 (84—48) — 2X48X27 


84+48 


= —23-64. 


the negative sign denotes that the body will return after the collision 
with a velocity of 23-63 feet per second. 

Example 3. Fig. 196. The partly elastic body M— 38 pounds and F=79 
ffeet per second, will strike the body in rest m =24 pounds ; what will be 
the velocity v'—1 of the body m, its elasticity being k'= 0-6- 


t/= 


79X38 (1+0-6) 


70*6 feet per second. 


38+24 

When a moving body strikes a stationary elastic plane, its course of 
departure from the plane will be equal to its course of incident. 



A 



■id. 


Problem. A body in a is to strike the plane 
AB so that it will depart to the given point b ; 
required its course of incident from a? 

Draw bd, at right angles through AB, make 
cd=*bc join a and d ; then ad is the course of in> 
cident, and eb, the course of departure, and the 
body will strike in e. 


























































824 


Centre of Gravity. 


I 


CENTRE OF PERCUSSION. 

Centre of Percussion is a point in which the momentuins of a moving 
body are concentrated. Centre of Percussion is the same as centre of oscillation , 
and to be calculated by the same formulas. 

Take an iron bar in one hand, and strike heavily over a sharp edge, if the 
centre of percussion of the bar strikes over the edge, the whole momentum will 
there be discharged, but if it strikes at a distance from the centre of percussion a 
part of the momentum will be discharged in the hand, and a shock felt. 

It is sometimes of great importance to properly place the centre of percussion 
If it is dislocated, the moving body not only fails to properly transmit its effect, 
but the lost momentum acts to wear out the machinery. 


♦♦ 


CENTRE OF GRAVITY. 

Centre of Gravity is a point around which the momentums of all matters 
(under the action of the force of gravity) in a body, or system of bodies, are 
equally divided. 



A body or system of bodies suspended at its centre of 
gravity, will be in equilibrium in all positions. 

A body or system of bodies, suspended in a point out of 
its centre of gravity, will hang with its centre of gravity ver¬ 
tical under the point of suspension. 

A body or system of bodies suspended in a point out of 
its centre of gravity, and having two different positions, 
the two vertical lines through the point of suspension 
will meet in the centre of gravity ; thus if a plane be hung 
up in two different positions, the vertical lines a, b, and 
c, d, will meet in the centre of gravity o. 

z = distance to the centre of gravity as noted in the 
figures. 

Example 1. The radius of a circle being 3 feet, how far is 
its centre of gravity from the centre of the half circle ? 

z = 0-6367 X3 = 1*91 feet. 

Example 2. How far from the bottom of a cylindric shell, 
open at one end, is its centre of gravity ? The cylinder is 
4 feet long, radius r = 0-8 feet. 


z 


h _ 4 

r+2h 0-8+2X4 


= 0‘625 feet. 


Example 3. Fig. 264. An irregular figure weighing P — 138 pounds, is sus¬ 
pended between a fulcrum and a weight, l = 5-6 feet, W— 57 pounds. Re¬ 
quired the distance to the centre of gravity z = ? 


57X5-6 

138 


2-31 feet. 




















Centre op Gravity. 


R'Vy 


a 

b 

252. 

Quadrangle.—a and b ‘parallel, 

h h ( b — as 

Z 2 6 'b+a > 


253. 

Triangle, 

h 

z= r 

.‘ 

254. 

Half a circle plane or Elliptic plane, 

z - 0*424r. 


255. 

Circle sector, 

2c r 

*"“ST 


256. 

Circle Segment, a = area. 

c 8 

* _ 12a' 

x = A+z — r. 

/tk 

257. Parabola. 

2 h 
•~5- 

3 

For half a Parabola .r = g- A. 



























Centre of Gravity. 


i 258. 



Half Sphere. 


Convex surface 
Solid . . . . 


z = £r. 
z — §r. 


259. 



—--A- 


Spherical Sector. 

Solid, 



260. Spherical Segment. 

Convex surface z = 



Solid 


A_ r2r*+#* -i 

2 l3r s +A* J 


261. 

Cone. 

Convex surface z = 

Solid 

** 

. II 

►H > 

262. 

Conic Frustum. 

h hrR-r-] 

Con. sur. z = 2 g-[ R+r -J 

Solid 

h r/2 a +r(2i2+3r) 1 

* 4 * L i2*+r(i2+r) J 

263. 

Fyramidic Frustum. 

A and 

a = area of the two bases. 

Solid 

^ ^ |~A+3a+2\/.A a 1 

4 l A+a^\/ A a J 



« z * 


< z 



< h > 





























Centre of Gravity. 


327 


k- l • 

a 

264. 

Irregular Figure . 

P : W = J: *, 

Wl 

■ z -~p- 



265. 

To the Centre of Gravity of two 

bodies , P and Q. 

Qfl , Pa 

* = P+Q ’ _ P+Q‘ 

T<t 

266. 

To the Centre of Gravity of a sys - 

tem of bodies. 

7. _ Fa ~ _ Q d 

b P+R’ P+R+Q ' 

G± 

\ 

267. 

Half a circumference of a Circle or 
Ellipse. 

z = 0-6367r. 

/^~r\ 

268. 

Circle arc or Elliptic arc. 

c r c(c a +4A 3 ) 

■ 

Z ~ b ~ Shb • 

< h -> 

W-M >•- 

i 

A. 

V 

a v 

269. 

Cylindric Surface with a bottom in one 
end. 

h 

x ~ r+2A' 































Specific- Grayitt. 


379 


^jfJECIFIC GRAVITY. 

Specific Gravity is the comparative density of substances. The unit for 
measuring the specific gravity is assumed to be the density of rain water, or 
distilled water. 

One cubic foot of distilled water weighs 1000 ounces, or 62-5 pounds avoir¬ 
dupois. 

To Find tile Weight of a Body. 

RULE 1. Multiply the contents of the body in cubic feet by 62-5, and the 
product by its specific gravity, will be the weight of the body in pounds 
avoirdupois. 

RULE 2. Multiply the contents of the body in cubic inches by 0-03616, 
and the product by its specific gravity, will be the weight of the body in 
pounds avoirdupois. 

RULE 3. Divide the specific gravity by 0-016 and the quotient is the weight 
of a cubic foot. 

Example 1. A bottle full of mercury is 3 inches, inside diameter, and 6 inches 
high. How much mercury is there in the bottle in pounds? 

One cubic inch of mercury weighs 0-491 pounds, and by the formula for 
Fig. 119 we have the 

weight = 0-49lX0'785X3 2 X6 = 20-85 pounds. 

Example 2. Required the weight of a cone of cast iron, diameter at the 
base d = 1-33 feet, height h = 4 feet? One cubic foot of cast iron weighs 
450-5 pounds, and by formula for Fig. 117 we have the 

weight = 450-5X0"2616Xl'33 a X4 = 834 pounds. 

Example 3. The section area of the lower hole in a steam boat is 245 square 
feet; how much space must be taken in the length of the hole for 131 tons 
of anthracite coal? 

Anthracite coal are 42-3 cubic feet per ton. 


length: 


42-3X131 

245 


= 22-6 feet, the space required. 


Weiglit and Bulk, of Substances* 




Cubic 

Oubic\ 



Cubic 

Cubic 

Names of Substances. 

foot 

in 

feet 

per 

Names of Substances. 

foot 

in 

feet 

per 



pounds. 

ton. 



pounds. 

ton. 

Cast iron, 
Wrought iron, 


450-5 

4-97 

Sand, 


94-5 

23-7 


486-6 

4-60 

Granite, * 


165 

13-5 

Steel, 


489-8 

4-57 

Earth, loose, • 


78-6 

28-5 

Copper, - 


555* 

4-03 

Water, salt, (sea) 


64-3 

34-8 

Lead, 


707-7 

3-16 

“ fresh - 


62-5 

35-9 

Brass, 


537-7 

4-16 

Ice, ... 


58-08 

38-56 

Tin,... 


456 

4-91 

Gold, 


1013 

2-21 

Pine, white 


29-56 

75-6 

Silver, 


551 

4-07 

“ yellow, - 


83-81 

66-2 

Coal, Anthracite 


53 

42-3 

Mahogany, 


66-4 

33-8 

“ Bituminous 


50 

44-8 

Marble, common, 


165 

13-6 

“ Ciunberland 


53 

42-3 

Mill-stone, 


130 

17-2 

“ Charcoal 


18-2 

123 

Oak, live - 


70 

32-0 

Coke, Midlothian 


32-70 

68-5 

“ white, 


45.2 

49-5 

“ Cumberland 


31-57 

70-9 

Clay, 


101-3 

22-1 

“ Natural Yirginia 

46-64 

48-3 

Cotton Bales, - 




Conventional rate of 



Brick, 


100 

22-4 

Stone coal, 28 bushels 



Plaster Paris, - 


105 

21-3 

(5 pecks) = 1 ton, 

- 


43-56 























Splcific Gravity. 


323 


To Find tlie Specific Gravity* 

W= weight of a body in the air. 

tv = weight of the body (heavier than water) immersed in water. 

S — specific gravity of the body. Then, 

W- w : W= 1 : & S = ' V -, .... 1 , 

IV — to’ ’ 

Example 4. Required the specific gravity of a piece of iron-ore weighing 
6*345 pounds in the air, and 4-935 pounds in water, S = ? 

S “ 6 ; 345 6 —4-935 = 4 ' 5 tho s P ecific gravity. 

When the body is lighter than water, annex to it a heavier body that is able 
to sink the lighter one. 

S — specific gravity of the heavier annexed body. 
s = specific gravity of the lighter body. 

W — weight of the two bodies in air. 
w = weight of the two bodies in water. 

V = weight of the heavier body in air. 
v = weight of the lighter body in air. 


V’ 

W — tv - 

8 


Example 5. To a piece of wood, which weighs v ■= 14 pounds in the air, is 
annexed a piece of cast-iron V — 28 pounds; the two bodies together weigh 
tu = 11-7 pounds in water. Required the specific gravity of the wood ? 

W = F+t> = 28+14 = 42 pounds. 

S = 7*2 specific gravity of cast-iron. 


Formula 2. 


5 = 


14 


42 —11-7 — 


= 0-529, the specific 


7-2 


gravity of the wood, (Poplar White Spanish.) 

A simple way to obtain the specific gravity of woods, is to form xt to a parallel 
rod, and place it vertically in water, then when in equilibrium, the immersed 
end is to the whole rod as the specific gravity is to 1. 

Example 6. A cylinder of wood is 6 feet, 3 inches long, when Immersed verti¬ 
cally in water it will sink 3 feet, 9 inches by its own weight. Required its spe¬ 
cific gravity. 

3-75 : 6-25 = S'. 1 , S= ^ = 0 - 600 . 

6-25 

To discover the Adulteration in Metals, or to find the proportions of two Ingredients 

in a Compound. 

„ W — s( W — w) 

V = _ j. 


5, 




Example 7. A metal compounded of silver and gold weighs W= 6 pounds 
in the air, and in water w — 5-636 pounds. Require tho proportions of silver 
and gold ? 

S = 19*36 specific gravity of gold. 
s = 10-51 specific gravity of silver. 

weight V = 6 ~ 5.636 ) _ pounds of gold. 

10-51 

1 19-36 and 1*246 pounds of silver. 









330 Specific Gravity. 




Weight 



Specific 

gravity. 

Weight 

Names of Substances. 

Specific 

gravity. 

per 

cubic 

Names of Substances. 

per 
] cubic 


inch. 




inch. 

Metals* 


lbs. 




lbs. 

Platinum, rolled - - 

22-669 

•798 

Alabaster, white 


2-730 

•0987 

“ wire, * - 

21-042 

•761 

“ yellow 


2-699 

•0974 

“ hammered, 

20-337 

•736 

Coral, red - - - 


2-700 

•0974 

“ purified, 

19-50 

•706 

Granite, Susquehanna 

2-704 

•0976 

* crude, grams 

15-602 

•565 

“ Quincy 


2-652 

•0958 

Gold, hammered - - 

19-361 

•700 

“ Patapsco 


2-640 

•0954 

“ pure cast - - - 

19-258 

•697 

“ Scotch - 


2-625 

•0948 

“ 22 carats fine - 

17-486 

•733 

Marble, white Italian 

2-708 

•097 S 

“ 20 “ - 

15-702 

•568 

“ common 


2-686 

*0968 

Mercury, solid at — 40° 

15-632 

•566 

Tale, black - • 


2-900 

•0105 

“ at+32° Fahr. 

13-619 

•493 

Quartz, - - - - 


2-660 

*0962 

“ « 60° “ 

13-580 

•491 

Slate, - - - - 


2-672 

•0965 

| « “ 212° “ 

13-375 

•484 

Pearl, oriental - 


2-650 

•095> 

Lead, pure .... 

11-330 

•410 

Shale, - - - - 


2-600 

•0940 

“ hammered - - 

11-388 

•412 

Flint, white - - 


2-594 

•0936 

Silver, hammered - - 

10-511 

•381 

“ black - - 


2-582 

•0933 

‘ “ pure .... 

10-474 

•379 

Stone, common - 


2-520 

•0910 

Bismuth, ----- 

9-823 

•355 

“ Bristol • 


2-510 

•0906 

Red Lead, ----- 

8-940 

•324 

“ Mill - - 


2-484 

•0897 

Cinnabar, ----- 

8-098 

•293 

“ Paving - 


2-416 

•0873 

Manganese, - - - - 

8-030 

•290 

Gypsum, opaque 


2-168 

•0783 

Copper, wire and rolled 

8-878 

•321 

Grindstone, - - 


2-143 

•0775 

“ pure - - - - 

8-788 

•318 

Salt, common - 


2-130 

•0770 

Bronze, gun metal 

8-700 

•315 

Saltpetre, - - • 


2-090 

•0755 

Brass, common - - - 

7-820 

•282 

Sulphur, native 


2-033 

•0735 

Steel, cast steel - - - 

7-919 

•286 

Common soil, - 


1-984 

•0717 

“ common soft - 

7-833 

•283 

Rotten stone, 


1-981 

0416 

“ hardened & temp. 

7-818 

•283 

Clay, .... 


1-930 

•0698 

Iron, pure - - - - 

7-768 

•281 

Brick, - - - • 


1-900 

•0686 

“ wrought and rolled 

7-780 

•282 

Nitre, - - * - 


1-900 

•0636 

“ hammered - - 

7-789 

•282 

Plaster Paris, - 

. r 

1-872 

•0677 

“ cast-iron - - - 

7-207 

•261 

1 

2-473 

•0894 

Tin, from Bohmen 

7-312 

•265 

Ivory, --- - 


1-822 

•0659 

“ English - - - - 

7-291 

•264 

Sand, - - - - 

- . 

1-800 

•0651 

Zinc, rolled - - - - 

7-191 

•260 

Phosphorus, - - 

- - 

1-770 

•0640 

“ cast ----- 

6-861 

•248 

Borax, - - - - 


1-714 

•0620 

Antimony, - - - - 
Aluminium - - - - 

6-712 

2-5 

•244 

0-09 

Coal, Anthracite 

- { 

1-640 

1-436' 

•0593 

•0592 

Arsenic, ----- 

5"763 

•208 

“ Maryland - 


1-355 

•0490 

Stones and Earths* 



“ Scotch - - 


1-300 

•0470 



“ New Castle 


1-270 

•0460 

Topaz, oriental - - 

4-011 

•145 

“ Bituminous 


1-270 

•0460 

Emery, ------ 

4-000 

•144 

Charcoal, triturated - 

1-380 

•0500 

Diamond, ----- 

3-521 

•127 

Earth, loose - - 


1-500 

•0542 

Limestone, green - - 

3-180 

•115 

Amber, - - - - 


1-078 

‘0387 

“ white - - 

3-156 

•114 

Pimstone, - - 


1-647 

•0596 

Asbestos, starry - • 

3-073 

•111 

Lime, quick - - 


0-804 

•0291 

Glass, flint .... 

2-933 

•106 

Charcoal, - - - 


0-441 

•0160 

“ white - - - - 

2-892 

•104 


“ bottle - - - - 

2-732 

•09S7 

Woods (Dry.) 



“ green - - - - 

2-042 

•0954 

Alder, - - - - 


•800 

•0289 

Marble, Parian - - - 

2-838 

•103 

Apple-tree, - - 


•793 

•0287 

“ African - - 

2-708 

•0978 

Ash, the trunk - 


•845 

•0306 

“ Egyptian - - 

2-668 

•0964 

Bay-tree, - - - 


•822 

•0297 

Mica, ----.. 

2-800 

•1000 

Beech, - - - - 


•852 

•0308 

Hone, white razor 

2-S38 

•104 

Box, French - - 


•912 

•0330 

Chalk, ------ 

2-784 

•100 

“ Dutch - - 


1-328 

•0480 

Porphyry, . 

2-765 

•0999 

“ Brazilian red - - 

1-031 

•0373 

Spar, green - - - - 

2-704 

•0976 

Cedar, wild - - 


•596 

•0219 

“ blue ... - 

2-693 

•0971 

“ Palestine 


•613 

0222 


















































Specific Gravity. 


331 


Names of Substances. 


Cedar, Indian - 
“ American 
Citron, 

Cocoa-wood, 
Cherry-tree, * 

Cork, 

Cypress, Spanish 
Ebony, American 
“ Indian 
Elder-tree, 

Elm, trunk of - 
Filbert-tree, 

Fir, male - 
“ female 
Hazel, 

Jasmine, Spanish 
Juniper-tree, - 
Lemon-tree, 
Lignum-vitse, * 
Linden-tree, - 
Log-wood, 

Mastic-tree 
Mahogany, 

Maple, 

Medlar, - 
Mulberry 

Oak, heart of, 60 ol( 
Orange-tree, - 
Pear-tree, 
Pomegranate-tree, 
Poplar, 

“ white Spanish 
Plum-tree, 
Quince-tree, - 
Sassafras, 

Spru e, - 
“ old 
Pine, yellow - 
“ white 
Vine, 

Walnut, - 
Yew, Dutch 
“ Spanish - 
Liquids. 
Acid, Acetic - 
“ Nitric 
“ Sulphuric 
“ Muriatic 
“ Fluoric - 
“ Phosphoric 
Alcohol, commercial 
“ pure 
Ammoniac, liquid 
Beer, lager 
Champagne, - 
Cider, 

Ether, sulphuric 
Egg, - 
Honey, - 
Human blood 
Milk. 


Specific , 

Weight 

per 

gravity. 

cubic 

1-315 

inch. 

•0476 

•561 

•0203 

•726 

•0263 

1-040 

•0376 

•715 

•0259 

•240 

•0087 

•644 

•0233 

1*331 

•0481 

1-209 

•0437 

•695 

•0252 

•671 

•0243 

•600 

•0217 

•550 

•0199 

•498 

•0180 

•600 

•0217 

•770 

•0279 

•556 

•0201 

•703 

•0254 

1-333 

•0482 

•604 

•0219 

•913 

■0331 

•849 

•0307 

1-063 

•0385 

•750 

•0271 

•944 

•0342 

•897 

•0324 

1-170 

•0423 

•705 

.0255 

•661 

•0239 

1-354 

•0490 

•383 

•0138 

•529 

•0191 

•785 

•0284 

•705 

•0255 

•482 

•0174 

•500 

•0181 

•460 

•0166 

•660 

•0239 

•554 

•0200 

1-327 

•0480 

•671 

•0243 

•788 

•0285 

•807 

•0292 

1-062 

•0384 

1-217 

•0440 

1-841 

•0666 

1-200 

•0434 

1-500 

•0542 

1-558 

•0563 

•833 

•0301 

•792 

•0287 

•897 

•0324 

1-034 

•0374 

9-97 

•0360 

1-018 

•0361 

•739 

•0267 

1-090 

•0394 

1-450 

•0524 

1-054 

•0381 

1-032 

1 *0373 


Names of Substances. 


Oil, Linseed • 
Olive - 
Turpentine 
Whale 
Proof Spirit, - 
Vinegar, - 
Water, distilled 
Sea 

“ Dead sea 
Wine, 

“ Port 


Miscellaneous. 


Asphaltum, 

Beeswax, - 
Butter, 

Camphor, 

India rubber, 

Fat of Beef, 

“ Hogs, 

“ Mutton, 
Gamboge, 
Gunpowder, loose 
“ shaken 

“ solid 

Gum Arabic, 
Indigo, - 
Lard, 

Mastic, 

Spermaceti, 

Sugar, 

Tallow, sheep 
“ calf 
“ ox, 

Atmospheric air, 


Gases* Vapours. 

Atmospheric air, - 
Ammoniacal gas, - 
Carbonic acid, - 
Carbonic oxid, 
Carburetted hydrogen, 
Chlorine, - 

Chlorocarbonous acid, 
Chloroprussic acid, 
Fluoboric acid, 
Hydriodic acid, 
Hydrogen, 

Oxygen, - 

Sulphuretted hydrogen, 
Nitrogen, 

Vapour of Alcohol, 

“ turpen’e spir., 
“ water, 

Smoke of bitumin. coal, 
“ wood, 

Steam at 212° - 


Weight 
Specific) per 
gravity, cubic 
inch. 


'940 

•915 

•870 

•932 

•925 

1-080 

1-000 

1-030 

1-240 

•992 

•997 


•905 

1-650 

•965 

•942 

•988 

•933 

•923 

•936 

•923 

1-222 

•900 

1-000 

1-550 

1-800 

1-452 

1-009 

•947 

1-074 

•943 

1-605 

•924 

•934 

•923 

•0012 


1-000 

•500 

1- 527 
•972 
•972 

2- 500 

3- 472 

2-152 

2-371 

4- 346 
•069 

1-104 

1-777 

•972 

1-613 

5- 013 
•623 
•102 
•90 
•4S8 


•0340 

•0331 

•0314 

•0337 

.•0334 

•0390 

•0361 

•0371 

•0448 

•0359 

•0361 


•0327 
0597 
‘0349 
‘0341 
*0357 
0338 
*0334 
’0338 
‘0334 
*0442 
•0325 
•0361 
•0561 
*0650 
•0525 
•0365 
•0343 
•0388 
•0341 
•0580 
•0334 
•0338 
•0334 
.43 

Weight 

cub. ft. 

grains. 

527-0 

263-7 

805-3 

512-7 

512-7 

1316 

1828 

1134 

1250 

2290 

36-33 

581-8 

9370 

512-0 

851-0 

2642 

328-0 

53-80 

474-0 

257-3 





























332 


Alloys. 


ALLOYS. 

A = Antimony, 2? — Bismuth, (7= Copper, G = Gold, I — Iron, L — Lead, 
N= Nickel, S = Silver, T— Tin, and Z — Zinc. 


Name. 

Brass, common yellow, 
Brass, to be rolled, 
Brass castings, com., 

“ “ hard,. 

Brass propellers, . 
Gun-metal,. 
Copper-flanges, 

Muntz’s metal, . . 

Statuary, 

German Silver, . . 

Britannia metal, . 
Chinese Silver, . . 

Chi. wht. Copper, . 

Medals, . . . 

Pinchbeck, . . 

Babbitt’s metal,. 

Bell metal, large, . 

“ “ small, 

Chinese gongs, 
Telescope mirrors, 
White metal, ord., . 

“ “ hard, 

Sheeting metal, 

Metal, expand in cool- 

iug, .... 


Alloy. 

20, IZ. 

32(7,10Z, 1.5 7’. 
206’ 1.25 Z, 2.52’. 
25 C, 2Z, 4.5 T. 
8(7,0.5Z,17’. 

SC,IT. 

9(7, IZ, 0.267’. 

6C, 4Z. 

91.4(7,5.53Z, 1.7 T, 
1.37 L. 

2(7, 7.9 N, 6.3 Z, 
6.51. 

50 A,25T,25B. 

65.1 C, 19.3Z, 132V, 
2.5SN, 127. 

20.2 (7,12.7 Z, 1.3 T, 
15.8 N. 
iooc;8Z. 

5(7, IZ. 

25T, 2A, 0.5 C. 

3(7, IT. 

4(7,1 T. 
40.5(7.9.271 
33.3(7,16.7 71 
3.7(7, 3.7 Z, 14.271 
28.4A. 

35(7,13Z, 2.271 
56(7,45Z, 12 arse¬ 
nic. 

75A,16.7A, 8.32?. 


Imitation of Gold. 

Melt separately, . j y Z 02 (7’ 9^* 
Gold imitation. . 71y, 9«. 


Type Metals. 

Name. Alloy. 

Smallest type, . 3A, 1A. 

Small type, . . 4A, 1A. 

Medium type,. . 5A, 1A. 

Large type, . . 6A, 1A. 

Largest type, . . 7 L, 1 A. 

Metal which can be 
forged at red heat, 
and strong as good 
iron, . . . 38.2Z, 60(7,1.757. 


Alloys for Solders. 


Name. Alloy. Melts. 

Newton’s fusible, 82?, 5L, ZT, 212° 

Rose’s “ 22?, \L,1T, 201° 

A more “ 52?, 3A, 271 199° 

Still more “ 1271 25A, 502?, 

13 cadium, 155° 

Por tin solder, 

coarse, . . IT 7 ,3 A, 500° 

For tin solder, or¬ 
dinary,. . 2T,\L, 360° 

For brass, soft spel¬ 
ter, . . . 1(7, IZ, 550° 

Hard, for iron, 2(7, IZ, 700° 


For steel, . 19N3C51Z. 

For fine brass work, IN, 8(7, 8Z. 
Pewterer’s soft sol¬ 
der, . . . 22?, 4 A, 371 

Pewterer’s soft sol¬ 
der, . . 12?, 1A, 271 

Gold solder, . 24(7,2^1(7. 
Silver solder, hard, 4N, 1(7. 

“ “ soft, 2 S, 1 brass wire. 

For Lead, . 167133A. 


Tempering of Steel. 

The property of heat to color steel or iron can be applied for ascertaining the 
temperature in flues and chimneys of steam-boilers, and for other temperatures 
limited between 430° and 600° Fah. 


Yellow, very faint, for lancets, . . . . 

“ pale straw, for razors, scalpels, . . . 

“ full, for penknives and chisels for cast iron, 

Brown, for scissors and chisels for wrought iron, . 
Red, for carpenters’ tools in general, 

Purple, for fine watch-springs aud table-knives, . 
Blue, bright, for swords, lock-springs, . 

“ full, for daggers, fine saws, needles, . 

“ dark, for common saws,. 


430° 
460° 
470° 
490° 
610° 
630° 
550° 
560° 
600° ' 






Relative Hardness of Substances. 


335 


Relative Hardness, II, of Substances. 


Minerals. 

H. 

Metals. 

H. 

Woods, Dry. 

H. 

Diamond. Ormuz, 

100 

Cast steel, hardened, 

65 

Chonta, S. Am., . 

28 

Diamond, Pink, 

97 

Cast steel, unhard., 

40 

Lignum vitae, . 

25 

Diamond, Yellow, 

94 

Cast iron, . 

38 

Ebony, . 

24 

Diamond, Cubic, 

92 

Iron, hammered, . 

37 

Pomegranate, . 

23 

Sapphire, 

90 

Pure iron,. 

35 

Boxwood, 

22 

Topaz, 

80 

Antimony, ham., . 

36 

Oak, very old, . 

22 

Garnet, . 

72 

Antimony, cast, 

32 

Oak, ordinary, 

21 

Agate, 

71 

Platinum, cast, 

40 

Mulberry, . 

20 

Amethyst. . 

71 

Platinum, ham., 

45 

Cedar, India, 

20 

Quartz, 

70 

Brass, common 

32 

Beech, 

19 

Ruby, pal a - Brazil, 

65 

White metal, hard, 

38 

Ash, 

18 

Ruby, 

64 

Gold, hammered, . 

30 

Alder, . . 

18 

Iron pyritt"), 

63 

Gold, cast,. 

26 

Apple tree, . 

17 

Opal, . 

62 

Copper, ham., 

34 

Plum tree, 

16 

Felspar, 

60 

Copper, cast, 

29 

Yew, 

15 

Fluor spar, 

40 

Silver, ham.,. 

32 

Maple, 

14 

Copper pyrites, . 

38 

Silver, cast, 

27 

Pine, yellow, 

14 

Calcareous spar, 

30 

Zinc, 

26 

Hazel, 

13 

Anthracite coal, . 

28 

Aluminum, 

24 

Cedar, wild, . 

13 

Galena, . 

27 

Tin, ham., 

24 

Birch, 

12 

Amber, . . , 

23 

Tin, cast, . 

20 

Fir, 

12 

Granite, . . 

22 

Babbitt’s metal, . 

20 

Pine, white, 

11 

Gypsum, 

20 

Silenium, . 

22 

Spruce, . . . 

10 

Bituminous coal, 

16 

Bismuth, 

20 

Sassafras, . . 

9 

Chalk, . 

15 

Lead, ham., . 

18 

Hemlock, 

8 

Talc, . 

10 

Lead, cast, 

15 

Cork,. 

5 


Mr. Chapman has arranged a scale for the hardness of minerals, as follows: 

1 yields easily to the nail. 

55 yields with difficulty .to the nail, or merely receives an impression from it. 
Does not scratch a copper coin. 

3 scratches a copper coin, but is also scratched by it, being of about the same 

hardness. 

4 not scratched by a copper coin. Does not scratch glass. 

5 scratches glass, though rather with difficulty, leaving its powder on it. Yields 

easily to the knife. 

6 scratches glass easily. Yields with difficulty to the knife. 

7 does not yield to the knife. Yields to the edge of a file, though with difficulty. 
8, 9 and 10, harder than flint. 

The numbers in Chapman’s scale multiplied by 10 will correspond with the 
hardness in the preceding table. 


Charcoal from 1000 Weights of Dry Wood. 


Oak, . 

. 226 

Beech, 

200 

Ash, . . .179 

Chestnut, 

. 232 

Fir, 

156 

Norwegian Pine, 192 

Mahogany, 

. 254 

Cedar, 

198 

Sallow, . . 184 

Walnut, 

. 206 

Pine, 

200 

Birch, . . 174 

Elm, . 

. 195 

Scotch Pine, 

164 

Sycamore, . . 197 























334 


Hydrometer. 


HYDROMETER. 

A body wholly immersed in a liquid will lose as much of its weight, as the 
weight of the liquid it displaces. 

A floating body will displace its own weight of the 
liquid in which it floats. 

A cylindrical rod of wood or some light materials, 
being set down in two liquids, A and B, of different 
specific gravities, when in equilibrium it will sink to 
the mark a in the liquid A, and to b in the liquid B; 
then the specific gravity of A : B — b, c : a, c, or in¬ 
verse as the immersed part of the rod. This is the 
principle upon which a hydrometer is constructed. 


270. 


n 


A 


-Li c 


r, 

~ f ^ 


a 


o b 
B=- 


Table showing the comparative Scales of Gay Lussac and Baumc, with the Specific 
Gravity and Proof at the temperature of 60° Fahr. 


271. 


F) 

Gay Lussac’s. 

Baumc $ 

Specific Grav. 

Proof. 

[- JCO 






90 

100 

46 

•796 

100' 


- na 

95 

40 

•815 

92 

F-i 

© 

-70 

-eo 


■3 90 

36 

•833 

82 

k 


•g 85 

33 

•848 

72 

© . 


8 80 

31 

•863 

62 

. cj O 


* 75 

28 

•876 

52 

+2 o 

-40 

£ 70 
g, 65 

26 

•889 

•901 

42 

a> 2* 


p 

24 

32 

© 

t-t 



'S 60 

23 

•912 

22 

Ah 

- =r^ 


° 55 

21 

•923 

12 


rb - —: 


m 50 

19 

•933 

0 Proof. 



a 45 

18 

•942 

8 1 


JpL 


S 40 
n 35 

17 

16 

•951 

•958 

18 

29 

bi. 4-5 
® O 
Tj o 



& 30 

15 

•964 

35 

P< 

p pm 



25 

14 

■970 

48 

"— 








HYDROSTATICS. 

Letters denote. 

A and a = areas of the pressed surfaces in square feet. 

I and p — hydrostatic pressure in pounds. 

d — depth of the centre of gravity of A or a under the surface of the liquids 
In fret. 

5 = specific gravity of the liquid. 

Example 1. Fig. 272. The plane A = 3-3 square feet, at a depth of d = 6 feet 
under the surface of fresh water. Required the pressure P = ? Specific gravity 
of fresh water S — 1. 

P = 62-5 A d = 62-5X3-3X6 = 1237-5 pounds. 

Example 2. Fig. 275. The area of the pistons A = 8-5 square feet, a = 0 02 
square feet, l = 4 feet, e = 9 inches, and .F= 18 pounds. Required the pres¬ 
sure P — ? 


P = 


FI A 18X4X8-5 


•e a 


= 40800 pounds. 


0-75X0’02 

It must be distinguished that the centre of pressure and centre of gravity of 
the planes, are two different points; the centre of pressure is below the ?entro 
of gravity, when the plane is inclined or vertical. 






























































Hydrostatics. 


835 






272. 

P = 62-5 SAd, 


■A- P 

62-5 S<2, 

- 

A. P 

62-5 S A. 

273. 

The Hydrostatic paradox . 

The pressure P is independent of the 
width of column C. 

P = 

62-5 S A A. (same as above.) 



P = a(62-5SA + 

p = a (:?-62-5 Sh), 

. P a -pA 
W5 SA a 

275. Bramah's Hydraulic Press. 
FI A A — - P ea 

PA l 


P = 


Pea 


P e 



276. Centre of Pressure of a rectangle , 
the upper edge at the surface 

of the liquid d — § A. 

277. Centre of Pressure of a triangle , 
the base bernu at the surface of 

the liquid , d *=\h. 

278. Centre of Pressure of a 
triangle, the vertex being at the surface 

of the liquid, d = 3 A. 

279. 


d = - + vA4 ( A — A') 3 + A*. 

























































Hydraulics. 


*36 


HYDRAULICS. 


Let the vessel A, Fig. 284, be kept constantly full of water up to the water 
line w. In two horizontal faces lower than the water line w, are made orifices 
a and a', through which the water will pass up vertical nearly to the water 
line w. Omitting the resistance of air, &c., the jet should theoretically reach 
the water line w, practically it reaches 0-967h. 

Jt is evident that the velocity of the jet thrbugh the orifices, must be the ve¬ 
locity due to a body falling the height h, according to the law of force of 
gravity. 

Letters denote. 

Q = actual quantity of water discharged per second or in the time t, in cubic 
feet. 

h — head, or height of water over the orifice. 
t = operating time in seconds. 
a = area of the orifice in square feet. 
m = the coefficient for contraction. (See Fig. 299 ) 

G = gallon of 231 cubic inches discharged in the time t. 

V — velocity through the orifice in feet per second. 

Example 1. Fig. 284. How many gallons of water will be discharged in five min¬ 
utes, through an orifice of 0'025 square feet, applied at 8 feet under the level of 
the water ? 

G = 37-75 a t yh = 37-75X0-025X5X60 ^8 = 800 gallons. 

Fig. 285. The weight P can represent the weight of a column of water whose 
P h' 


height = 


62-54 0-967 


acting on the area A. 


Fig. 286 .n = number of down.strokes per minute, s = stroke of piston; the 
air vessel (7= 6 A s at the pressure of the atmosphere. 

Example 2. Fig.286. How many double strokes must be made per minute by 
the lever of a hand pump, to throw up 22 cubic feet of water 18 feet high, in the 
time of 8 minutes and 15 seconds; the levers l = 30 inches, e — 8 inches, 
s — 0"o feet, F = 20 pounds ? 8X60+15 = 495 seconds. 


n = 


3630 Q h' e 3630X22X18X8 
tsFl = 495X0-6X20X30 


= 64-5 strokes per minute. 


Example 3. Fig. 294. A vessel of rectangular form is of dimensions A — 0 
square feet, the height h == 5 feet. What time will it take the water level to 
sink 2 feet, when the orifice a = 0-212 square feet. 


t = 


A (ft — AQ 


6(5 — 3) 


2-52a(yA+y7t/ 2-52X0-212(^5+v 7 3) 


= 5-66. 


Motion of Water in Pipes) 

Letters denote. 

L = extreme length of the pipe in feet. 

d = inside diameter in feet, and uniform throughout the length L. _ 

Example 4. Fig. 287. What will be the velocity of the water through a pipe of 
0 45 feet inside diameter, and L = 68 feet long, the head pressure of water being 
h =- 8 feet? 


'-“n/ 


0-45X8 

68+50X0-45 


= 9-6 feet per second. 
















22 























































338 


Hybraulics. 




290. 


Weirs . 

Q =. ]i b t. See Table for Weirs. 

,_Q h Q 

kb* b = kt* 


291. 


Q = 5-35m £ A t>J~K y 
G = 40m b h ty/ 

t _ _ 

1 ~ 5‘35 m A An/ A * 



292. 

Q = 5-35m 6 t(hVJT— A' 

G = 40m b /(Av/T— AVA 7 ), 

Q 


t = 


5*35m A(AV h—^hW A') * 



^ 0*95m4.(>/A— >/A r ) 

~b~7T7 * 

4 = area of the vessel in square feet. 
t = time in seconds, in which the water level 
will sink the space h — h'. 


4 _ A(h — A') 

L ~ 


4m a (y/ h+ y/ A')> 
Q = 4m a t(s/ A+ A'), 


_ 

_ A— 

h 

A 

V 

—— 

wzzzzzr^r- 



A 

t 


3 - 85a 


m 


(V*— v^ 7 ). 


Ay/h __ i 

a ~ 3-85 Tm * 


A sTh 
3-85 am 



































































Hydbmjlics. 


Sd'J 




















































































840 


Hydraulics. 


Example 5. Fig. 289. Required the velocity and quantity of water discharged 
in a long pipe or hose of L = 135 feet long, and d — 0.17 feet, attached to a hand- 
pump of D — 0.2 feet in diameter P = 44 pounds, and the end of the pipe elevated 
h = 200 feet above the piston D1 


V= 6.86 * °: 17 ( U — 49 X 0--" X ~ 0) _ 4.95 f ee t p Pr second. 

\ 0.2(135 + 50 X 0.17) 

Q = 1.95 X 5-38 X 0.22 = 0.042 per second X 60 = 2.52 cubic feet per minute, 
s = 0.8 feet, the stroke of piston, we shall have 

n —--= 100 strokes per minute. 

0.8 X 0.785 X 0.22 


Table for Water flowing over Weirs. 


This table is set up from careful experiments 
on a large scale, and is suited for weirs only. See 
Fig. 290. q = 4.327 b yw. 

Rule. Multiply the width b, in feet, of the 
weir by the coefficient Ic, and the product is the 
quantity of water discharged per second, in cubic 
feet, h is the height as represented by Fig. 290. 
The width b should be b > h. 

Example 6 . How much water will flow over a 
weir of b = 5 feet, h — 0.5 feet, in one minute ? 

Q = Jcb t = 1.1295 X 5 X 60 = 338.35 cubic feet. 


h. inches. 

h. feet. 

m. 

k. 

0.4 

0.033 

0.424 

0.01365 

0.8 

0.066 

0.417 

0.05452 

1.2 

0.100 

0.412 

0.10592 

1.6 

0.133 

0.407 

0.16616 

2.4 

0.200 

0.401 

0.29171 

3.2 

0.266 

0.397 

0.44480 

4. 

0.333 

0.395 

0.63111 

6. 

0.500 

0.393 

1.1295 

8. 

0.666 

0.390 

1.7464 

9. 

0.750 

0.385 

2.0331 

12. 

1.000 

0.376 

3.1350 


On tlie Velocity of Water in Rivers. 

Notation of Letters. 

F = fall of the river in feet per mile. 

R = hydraulic radius in feet, or the area of the cross-section of the river in 
square feet, divided by the wet perimeter in feet. 

V = mean velocity of the water in inches per second. 

M — mean velocity in miles per hour. 

F=10.9 VF~B, F= - F —. 

v ’ 118.8 R 


M= 0.619 l fFR, 


F= — L . 

3.83 E 


The mean velocity of the water throughout the whole section of the river is to 
the velocity at the surface in the middle of the river as 84:100, or as 100 : 120 . 


Example 1. The cross-section of a river is measured to be 560 square feet, and 
the wet perimeter 196 feet; the fall of the river is 5 feet per mile. Requii’ed, the 
hydraulic radius and the mean velocity of the water in miles per hour? 


Hydraulic radius R 


560 

196 


2.86 feet. 


Mean velocity M = 0.619 j/5 X 2 86 = 2.34 miles per hour. 

Example 2. The velocity of the surface in the middle of a river is 36 inches per 
second; hydraulic radius It = 2 feet. Required, the mean velocity and the fall of 
the river per mile? 


Mean Velocity V = 36 X 0-84 = 30.24 inches per second. 
30 24 2 

Fall F = - L ~- = 3.8487 feet per mile. 

118.8 X 2 F 




















Hydraulics. 


841 


Obstruction in Rivers. 

R — rise of water in feet caused by obstruction. 

A = sectional area in square feet of river unobstructed, and a = that when ob¬ 
structed. V — velocity in feet per second of the water without obstruction. 




Resistance to a Plane Facing a Current of Water 

or Moving in Still Water. 

A — area of the plane in square feet. 

R = resistance in pounds. 

V = velocity in feet per second. 

R = A V 2 , in fresh water. 

R = 1.032 A V 2 , in salt or sea water. 

When the plane is set at an angle of less than 90° to the direction of motion, the 
resistance will be, when (j> = angle of the plane, 

R = A (V sin.0) 2 , in fresh water. 

R — 1.032 A (V sin .<p) 2 } in salt water. 

Theoretical Velocity of Water, due to Head of Fall. 

See table for falling bodies, page 308, in which the column S represents the 
head of fall in feet. 

To find the lumber of Gallons of W T ater G which can be 
raised per Hour from a Well of Depth D, 

By a Suitable Double-action Force-and-lift-pump. 

D may also denote the height to which water may be raised in water-works. 

.. 18000 

One man working a crank, 


A donkey working a gin, 
A horse working a gin, 
Per steam horse-power, 


G = 
G 
G 


JD 

36000 

D 

126000 


G = 


D 

190000 


D 


or 0.8 of the natural effect. 

Example 1. How many gallons of water can be raised per hour from a well 
150 feet deep by a horse working a gin ? 

^ 126000 0 . A „ , 

G =-= 840 gallons, the answer. 

150 6 

Example 2. How many gallons of water can be raised per hour to a height of 
D — 150 feet by a steam-engine of 120 actual horse-power? 


G = 


190000 X 120 
150 


= 152,000 gallons, the answer. 










842 


Hydraulics. 


MOTION OF WATER IN PIPES. 

Letters denote — 

Q = cubic feet of water passed through the pipe per minute. 

D = inside diameter of the pipe in feet. 

L = length of the pipe in feet, increased by 50 diameters. 

H = head of fall in feet. 

V — velocity of the water in the pipe in feet per minute. 


Q = 2356 


Af 


ww 


D = 


22.329 V S’ 


Q*L 


V= 3000 


V 


HI) 
L ' 


Example 1. A water-pipe of D = 1.75 feet in diameter, L = 36,000 -f 50 X 1.75 
= 36087.5 feet long, head pressure U = 390 feet. Required, how much water it 
can discharge per minute? 


Q = 2356 J —° X 1 - 75 - 5 = 992.26. 
v \ 36087.5 


Example 2. At a distance of 27960 feet from a water-work is required Q — 564 
cubic feet of water per minute, head pressure being H — 256 feet. Required, the 
diameter of the pipe ? L = 27 960 + 50 = 28010 feet. 


D = 


156 41X28010 = 1 .4436 feet. 

\ 256 


22.329 

Example 3. A water-pipe of D = 0.75 feet in diameter, L = 8650 + 50 = 8700 
feet, has a head pressure of H= 128 feet. Required the velocity v = ? of the 
discharge. 


F=3000 -yj 


12 8 X 0.75 
8700 


= 315.13 feet per minute. 


January, 

2.58 

April, 

2.73 

July, 

4.58 

February, 

2.40 

May, 

3.37 

August, 

4.75 

March, 

2.64 

June, 

3.50 

Sept., 

4.61 


Consumption of Water in Cubic Feet per head of Population, 

Including all Uses, as for Manufactories , Fires, etc., in 24 hours. 

October, 4.46 
N ovember, 4.12 
December, 3.61 

On tlie Flow of Water in Bends of Pipes. 

Notation of Letters. 

L = the whole length of pipe in feet, straight and curved or bent, increased 
with 50 D. 

R = radius in feet of the bend of the centre-line of the pipe. 

<!> = angle of deflection or bend of pipe in degrees. Should the pipe have several 
bends, add all the angles to <f>. 

Sin. <j> to be used only up to 90°, and disappears in the formulae for greater 
angles. 

D — inside diameter in feet of the pipe; V = velocity of the water in feet per 
minute; and H — head of fall in feet; Q = cubic feet of water dis¬ 
charged per minute. 

Q = 2356 (i 4 - ^ Sm - n . 

v V i H90/1 T J B-fl0) 

V= 3000 J—( - 90 )fl + — 

V L \<1> + 90/ V ^ B+IOJ 

The formulae will answer for a pipe of the form of a screw-spiral. 





























The Hydraulic Ram. 


443 



THE HYDRAULIC RAM. 

This hydraulic motor appears to be too little known in many parts of the world. 
The author of this book has been in the interior of many countries where water 
is raised in a very rude and laborious way, and where the hydraulic ram would be 
of great utility. The useful effect of the ram, like that of water-wheels and tur¬ 
bines, depends much upon its construction. In ordinary cases it returns about 50 
per cent, of the natural effect. That is, the quantity of water ( q) multiplied by 
the height ( h ) of the delivery aliove the ram will be about 50 per cent, of the 
quantity of water ( Q) working the ram, multiplied by the head of fall (F), in the 
ame unit of time. 


qh = 0.5QF. 


9 = 


0.5 QF 

h * 


Q = 


2 qh 


F 


Q and q can be expressed in any unit of volume or weight. 

F and h can be expressed in any unit of length. 

But let us assume Q and q to be cubic feet per minute, 

F and h — fall and height in feet, 

L — length in feet and D — diameter in inches of the supply-pipe S t 
l = length and d = diameter of the delivery-pipe d. 


Then D 




W(L + 5D) 


and d = ^ («+■“). 


Description of tlie Hydraulic Ram. 

Reference to the figure above. 

The water working the ram is supplied through the pipe S, and escapes through 
an opening at o, until it has gained a velocity sufficient to raise the valve or ball 
B, which suddenly stops the current, and causes an excessive pressure in the ram 
jR, which opens the valve or ball G\ the water is forced into the vessel and air- 
chamber A, and finally through the delivery-pipe d to its destination. When 
equilibrium of pressure is restored between »!? and R, the ball B falls, and the 
operation is repeated. The ram can make as much as 200 strokes per minute, 
depending upon its size. 

The length of the supply-pipe S should not be less than five times the height 
of the fall F, because it is the dynamic momentum (see page 310) in the pipe 
columns of water which works the ram. But the pipe may be made 10 times, or 
more, the height of- the fall. 





















Hydrodynamics. 


344 


HYDRODYNAMICS. 

“Water Power. 

The natural effect concentrated in a fall of water is equal to the weight of the 
quantity of water passed through per second, multiplied by the vertical space it 
falls. 

Fig. 297. Let Q be the quantity of water which passes through the orifice a in 
the time t — 1" second, in cubic feet of 62.5 pounds each. 

h = the vertical space the water falls; then the value or natural effect of the 
fall is at the orifice a. 

P = 62.5 Q h, effects of power. 

Q — 5.06 a h\/h ; 

P == 315.5 a hy'h. 

This will be in horse power. H = 0.5/ 3 (X h\ / h t _ET= 0.1134(2 h, 

II 


But, 

Then we have 


—_JL *\E1 

_ 1.07 \ a 2 ’ 


h 


0.1134 Q 

Example 1. In a creek passes 18 cubic feet of water per second. How high 
must that creek be dammed up to produce an effect of ten horses ? 

h = -= 4.9 feet, the answer. 

0.1134 X 18 

Comparison of Columns of Water in Feet. 

Mercury in inches, and pressure in pounds, per square inch. 


Pounds 

Water. 

Merc’ry. 

Water. 

Merc’ry. 

Pounds. 

Merc’ry 

Water. 

Pounds. 

Pr.sq.in. 

Feet. 

Inches. 

Feet. 

Inches. 

Pr. sq. in. 

Inches. 

Feet.. 

Pr. sq. in. 

1 

2.311 

2.046 

1 

0.8853 

0.4327 

1 

1.1295 

0.4887 

2 

4.622 

4.092 

2 

1.7706 

0.8654 

2 

2.2590 

0.9775 

3 

6.933 

6.138 

3 

2.6560 

1.2981 

3 

3.3885 

1.4662 

4 

9.244 

8.184 

4 

3.5413 

1.7308 

4 

4.5181 

1.9550 

5 

11.555 

10.230 

5 

4.4266 

2.1635 

5 

5.6476 

2.4437 

6 

13.866 

12.2276 

6 

5.3120 

2.5962 

6 

6.7771 

2.9325 

7 

16.177 

14.322 

7 

6.1973 

3.0289 

7 

7.9066 

3.4212 

8 

18.488 

16.368 

8 

7.0826 

3.4616 

8 

9.0361 

3.9100 

9 

20.800 

18.414 

9 

7.9680 

3.8942 

9 

10.165 

4 39S7 

10 

23.111 

20.462 

10 

8.8533 

4.3273 

10 

11.295 

4.8875 

11 

25.422 

22.508 

11 

9.7386 

4.7600 

11 

12.424 

5.3762 

12 

27.733 

24.554 

12 

10.624 

5.1927 

12 

13.554 

5.8650 

13 

30.044 

26.600 

13 

11.509 

5.6255 

13 

14.683 

6.3537 

14 

32.355 

28.646 

14 

12.394 

6.0582 

14 

15.813 

6.8425 

15 

34.666 

30.692 

15 

13.280 

6.4909 

15 

16.942 

7.3312 

16 

36.977 

32.738 

16 

14.165 

6.9236 

16 

18.072 

7.8200 

17 

39.288 

34784 

17 

15.050 

7.3563 

17 

19.201 

8.3087 

18 

41.599 

36.830 

18 

15.936 

7.7890 

18 

20.331 

8.7975 

19 

43.910 

38.876 

19 

16.821 

8.2217 

19 

21.460 

9.2S62 

20 

46.221 

40.922 

20 

17.706 

8.6544 

20 

22.590 

9.7750 

21 

48.532 

42.968 

21 

18.591 

9.0871 

21 

23.719 

10.264 

22 

50.843 

45.014 

22 

19.477 

9.5198 

22 

24.849 

10.752 

23 

53.154 

47.060 

23 

20.362 

9 9525 

23 

25.978 

11.241 

24 

55.465 

49.106 

24 

21.247 

10.3S5 

24 

27.108 

11.7300 

25 

57.776 

51.152 

25 

22.133 

10.818 

25 . 

28.237 

12.219 

26 

60.087 

53.198 

26 

23.018 

11.251 

26 

29.367 

12.707 

27 

62.398 

55 244 

27 

23.903 

11.683 

27 

30.496 

13.196 

28 

64 709 

57.290 

28 

24.789 

12.116 

28 

31.626 

13.685 

29 

67.020 

59.336 

29 

25.674 

12.549 

20 

32.755 

14.174 

30 

69.331 

61.386 

30 

26.560 

12.981 

1 30 

1 33.885 

14.662 























Water-Wheels. 


345 


WATER-WHEELS. 

Water-wheels are of two essential kinds, namely, Vertical and Hcn'izemtal. 

The Vertical are subdivided into 

Overshot-wheels, Undershot-wheels, Breast-wheels, and High-breast and Low-breaft 
wheels. 

The Horizontal are with Floats , Screw-wheels, Turbine, Reaction-wheels, <£c. 

Waterwheels do not transmit in full the natural effect concentrated in a fall 
of water; under most favourable circumstances 80 per cent, has been utilized, 
but under poor arrangements only 20 per cent, may be expected. 

Example 1. Fig. 302. The vertical section of the immersed floats of an under¬ 
shot-wheel in a mid-stream is a = 27 square feet, velocity of the stream V— 8 - 6 , 
and v = 4 feet per second. Required the horse-power of the wheel H -- ? 

a v 27 V4 

H = r^( F ~ v ) a = l>^r( 8 ' 6 — 4 ) a = 11,4 hor ses. 

Example 2. Fig. 307. On a breast-wheel is acting Q — 88 cubic feet of water 
per second, the head h = 8 feet, velocity of the wheel at the centre of the 
buckets v — 5 feet p^r second; the water strikes the buckets at an angle u — 8 ° 
and velocity V = 7 feet per second. Required the horse-power of the wheel, 
H = ? 

88 / 5 \ 

II = pp 4 .\ 8 + 2 ^( 7 X c °s. 8 0 — 5) 1 = €5 horses. 

Example 3. Required the effect of Poncelet’s wheel, the head h = 4 feet, and 
the orifice a = 5 square feet, the velocity of the wheel at the centre of pressure 
of the floats is v = £’78 feet per second? 

V = 6-91 yi = 13-82 feet per second. 

Q = O^X^X^ = 65 cubic feet per second. 

H = (13-82 — 6-78) = 15‘8 horses. 

Example 4. Fig. 309. A saw-mill wheel is to be built under a fall of h = 18 
feet, and to make n = 110 revolutions per minute. Required the proper diam¬ 
eter of the wheel. 

D^ o yl 8 '= 3-857 feet, 

at the centre of pressure of the buckets. 

Felocity V = 8^18 — 33-94 feet per second. 

Velocity v= ————— = 22-2 feet per second. 

The fall discharged 30 cubic feet of water per second. Required the hone 
power of the wheel. H = ? 

H = C33-94 — 22-2) = 39 horses. 

How many square feet of dry Pine can it saw per hour ? 

See page 264. 30X39 =1170square feet. 

The saw is meant to be applied dire -t on the wheel shaft. 
















m DR A cue?. 


818 



























Hydraulics. 


347 






206. Low-breast Wheel. 

11 ~ T^“[ a + -S2~( v cosu - * )] 


Q = kb. V = -S • See table for weirs. 
a 


307. Breast Wheel. 


H - TT4 + 



308. 


Over-shot Wheel. 


f H = lM A + 3F5 ( 


Proper velocity about n = ^ r 


revolutions per minute. 


309. 


Saw-Mill Wheel. 


V-v) 

200 v 1 

Proper diameter of the Wheel , 

100 /T - . r, 

J) = - V h , m feet, 

n 

n = revolutions per min. 













































348 


Turbine Wheels. 


TURBINES. 

V 

Letters denote. 

Q cubic feet of water passed through the turbine per second. 
h = height of fall in feet. 

D = diameter in inches of circle of effort in the turbine. 
a= area in sq. in. of the conduit passage into the turbine wheel. 

6 = depth in inches of turbine buckets. 
c = depth in inches of leading buckets. 
r= breadth of turbine buckets in inches. 
m — number of buckets in the turbine wheel. 
m'~ number of leading buckets, 
n = number of revolutions of turbine per minute. 

S and s=height of conduit and discharge in inches. 
t = thickness of steel plate buckets in 16ths of an inch. 

H= actual horse power of the turbine. 

I = length in feet 1 f d nine 
d = diameter in inches j 01 conQmt P x P e - 
d' = diameter in inches of the discharge pipe. 

W= Hydraulic pressure on the turbine wheel bearing on the end oi the 
shaft. 


_ ky h 

2 > = “ 


n 


D— 


n 


n- 


0-436 r 

_ ky h 

~~W ’ 

20 k Q 
a D* 

a 

0-436H’ 


46 k Q 

r 

6 8 ’ 


- 2 


- - - 3 


- - 4 


t = 


m 


10 ’ 


- - 6 


- - 7 


- 8 


a = 


a = 


20 Q 

y h ’ 

20 k Q 


D n 
a — 0-436 Dr, 


- - 9 


- - 10 


11 

m'rs, - - - 12 
mrs,- - - 13 
14 


a' 

a'=0-98 a, 


Q 


_ a)/h 


20 


n a D n 
20 * ’ 


15 


16 


m = 5 ]/H, - 
w»'=4*5j/P, - 
0-626 D 


6 = 


y m 


- 17 


- 18 


- 19 


0-78 D 


y to ' ’ 
s = 0-86 S, 

d = D-)-r+ VX 
H+2 r, - - 

W™ . - 


- 20 


- - 21 

22 

23 

24 


H 


H= 0-1134 Q h 

30 Q 3 ") 
- zi—> ! 


H __ d 

267-6 ’ 


natural effect of the fall, 

actual horse power, 

66 per cent of the natural. 


25 

26 
27 


The coefficient k can vary from 800 to 1200 without seriously affecting 
the per centage of the ultilized power, but it is best between 900 and 1000. 
This is a great advantage of the turbine over water wheels, that under 
the same head of fall it can run at different velocities and still utilizing 
the maximum effect. Whatever coefficient k adopted it must be kept the 
same throughout the construction of the turbine. 

























Turbine Wheels. 


349 


" 

Jonval’s Turbine has so many advantages above other hydraulic mo¬ 
tors that it is considered sufficient to describe the construction of that 
one only, but the principal formulas will answer for any kind of turbines. 

On the accompanying plate is a drawing of a Jonval Turbine such as i 
the Author of this Pocket Book haB built in Russia. The buckets are not j 
supported by concentric rings, but are fastened only on one side, which J 
is considered more simple and convenient for replacing new buckets. For 
falls over 30 feet it may be better to make it with concentric rings. 
When a turbine is to be constructed we have on the one side given the 
natural effect of the fall, and on the other side the actual work to bo 
done, which latter should not exceed G6 per cent, of the former. Between 
these two points the turbine is to be so proportioned as to utilize the 
greatest possible effect with smallest expense of Machinery. 

Jonval’s turbine in good condition generally utilizes 60 to 80 per cent. 
Suppose a fall of ft 26 feet, discharging Q 12 cubic feet of water per 
second, the natural effect will be, 

J/ = 0-1134X12X 25 = 34 horses, 

of which 34X'66—22-4 horses to be counted upon as the actual effect of 
the turbine. 

Turbine shaft to make n=200 revolutions per minute with the assumed 
coefficient k 960. From these dates we will obtain all the principal 
dimensions of the turbine, namely, 


„ 960 y 25 . . 

D= -— ' - = 24 inches. - - 1 

200 


20X960X12 

a = -—— = 48 sq. in. 10 

24X200 1 

m = 5p 24=24-5 say 25. - - 17 

m' = 4-5j/24 = 22 buckets. - 18 


r = 


48 


= 4-6 in. - - - 6 


0-436X24 

0*5X84 

y 25 

0-78X24 , . . 

c = - — = 4 inches. 

y 22 


- - - 19 


- - 20 


t = 2 - = 2-5, 16ths. 
10 


- - - - 8 


In calculating the breadth r from formula 5, it must come Inside of 
formula 7, if not the diameter D must be altered. 

Now proceed with the construction as shown at the bottom of the plate, 
which represents a section of the buckets through the circle of effort of 
the turbine. 

The drawing of the turbine is J of an inch to the foot, and the construc¬ 
tion of the buckets 3 inches to the foot. 

Draw the base line A/J, set off the angle of the leading buckets 10°. 
The distance between the leading buckets will in this case be 24X3-14:22= 
3-43 inches, set off this from S towards A, draw the straight part of the 
second bucket parallel to the first one, draw from S the line d d at right 
angle to the buckets, and c will be the centre for the curved part. From 
the centre of S draw the line o to the end of the second buckets, divide 
this line into eight equal parts take five of them as radus ami draw from 
the end of the second bucket a circlearc of about 60°, which will be the 
pi-opelling part of the turbine wheel bucket. 

Distance between the wheel buckets will be 24X3-14:25=3-02 inches, set 
off this from A towards S, draw the second propelling arc. Set off from A 
the depth of the wheel buckets b 3 inches, set; off 2 b to s, which will be 
the length of the first wheel bucket. Set off from s to u the distance 
between the buckets 3-02 inches. Make s 0-86 S. Draw from u a curved 
line in the form of a parabola that will leave the space s and tangent the 
propelling circlearc somewhere about x. Care must be taken that the 
discharging area a' of all the wheel buckets will be about 2 per cent, less 
than the conduit area a of all the leading buckets. The surface of the 
buckets should be made as smooth as possible, or even polished. 

For very high falls the Hydraulic pressure W becomes very considerable! 
















350 


Turbines. 


and may necessitate another arrangement, namely, to lay the shaft horizontally 
and place on it two turbines, so that the leading buckets are either between or 
outside of the wheels; but then comes another disadvantage, namely, that the 
number of revolutions will be greatly increased and may be required to gear it 
down 10 to 20 times to the proper speed of the main shaft. 

To avoid this as much as possible, take Jc = 800, and make r — 

One great advantage with Jonval’s turbine is that it can be placed almost any¬ 
where between the high and low levels to suit the location, though it should not 
be more than 20 feet above the lower level; then, in order to utilize the whole fall, 
care must be taken to make the discharge-pipe perfectly air-tight. It is not neces¬ 
sary to make the discharge straight down from the turbine: it can be carried hori¬ 
zontally or inclined, as may suit the location. The author has built turbines 
similar to that represented on the accompanying plate, at General Maltzof’s es¬ 
tablishment, Kaluga, Russia. 

Approximate or Proportionate Price of Turbines, 

as fitted and delivered at the foundry, without shaftings or gearings, is — 

* 40 0^/H 

*Jp —--1 

tfF 

in which H— horse power of the turbine and F the height of fall in feet. 

Example. Required the price of a turbine, H = 100 horses, to work under a fall 
of F= 25 feet. 


$ = 4 -^° = =1375 dollars. 

^25 2 - 92 

Price Hist of Turbines in Dollars. 


Head of fall in feet, F. 


power. 

5 

10 

15 

20 

30 

40 

• 50 

75 

100 

150 

h 

$ 

$ 

% 

$ 

$ 

$ 

$ 

$ 

$ 

$ 

i 

234 

186 

163 

148 

130 

117 

110 

95 

86 

76 

2 

330 

263 

231 

209 

183 

167 

154 

134 

122 

107 

4 

467 

372 

326 

295 

258 

235 

218 

190 

172 

151 

6 

552 

455 

400 

262 

316 

288 

266 

232 

211 

185 

8 

660 

526 

462 

418 

365 

332 

308 

269 

244 

213 

12 

808 

642 

565 

510 

447 

405 

377 

329 

300 

261 

16 

935 

742 

654 

590 

516 

468 

434 

380 

345 

302 

20 

1045 

830 

730 

660 

577 

524 

485 

425 

385 

338 

30 

1280 

1020 

894 

810 

705 

642 

595 

520 

472 

414 

40 

1480 

1180 

1035 

932 

815 

740 

686 

600 

545 

476 

50 

1650 

1320 

1155 

1045 

913 

828 

7 68 

671 

610 

532 

60 

1810 

1440 

1264 

1140 

1000 

908 

840 

735 

668 

584 

80 

2090 

1645 

1460 

1320 

1155 

1050 

975 

848 

770 

674 

100 

2340 

1860 

1630 

1480 

1295 

1175 

1090 

948 

860 

753 

125 

2620 

2080 

1820 

1650 

1440 

1310 

1220 

1060 

984 

845 

150 

2860 

2280 

2000 

1810 

1580 

1440 

1330 

1170 

1060 

924 

175 

3100 

2460 

2150 

1950 

1700 

1550 

1440 

1260 

1140 

1000 

200 

3310 

2630 

2300 

2090 

1825 

1655 

1540 

1350 

1220 

1066 

225 

3510 

2790 

2440 

2215 

1935 

1755 

1630 

1430 

1300 

1130 

250 

3700 

2940 

2570 

2335 

2031 

1850 

1720 

1500 

1365 

1195 

275 

3890 

3090 

2700 

2450 

2135 

1945 

1800 

1580 

1430 

1255 

300 

4060 

3225 

2820 

2560 

2235 

2030 

1890 

1650 

1500 • 

1300 

350 

4380 

3480 

3035 

2760 

2410 

2190 

2035 

1780 

1610 

1410 

400 

4680 

3720 

3250 

2950 

2580 

2340 

2175 

1900 

1725 

1510 

450 

5080 

3950 

3450 

3130 

2740 

2480 

2310 

2015 

1825 

1600 

500 

5240 

4160 

3610 

3300 

2890 

2620 

2535 

2125 

1925 

1685 






























Mate II. 






















































































' 


. 























* 


- 










■ 





























. 


















'V 


... 







• 






































Coefficient for Capacity and Weight. 


3.11 


Coefficient for Capacity and Weight, 


1 



-■ 




MSBML J 


Names of Substances. 

FFF. 

Fix. 

Hi. 

FF*. 

Ft a. 

n». 

F*. 

t*. 

Cubic inches, - 


1728 

12 

1 

1356 

9-42 

0-785 

903-7 

0-523 

Cubic feet, - - 


1 

-..694 

•....58 

0-785 

*..549 

*.41 

0-523 

•.3 

Gallons, - - - 


7-476 

0-052 

•...433 

5-868 

-.408 

*.34 

3-91 

*..226 

Water, fresh, - 


62-5 

0-433 

0-036 

49 

0-34 

*. 283 

32-7 

0-019 

Water, salt, - - 


64-3 

0-445 

0-037 

' 50-4 

0-35 

0-029 

33-6 

0-02 

Oil,. 


57-5 

0-4 

0-033 

45-1 

0-313 

0-026 

30 

0-017 

Cast-iron, - - 


450 

312 

0-26 

353 

2-45 

0-204 

235 

0-136 

Wrought-iron, - 


487 

3-37 

0-281 

382 

2-65 

0-221 

255 

0-147 

Steel, - - - - 


490 

3-4 

0-283 

385 

2-67 

0-222 

257 

0-149 

Brass, .... 


532 

3-68 

0-307 

417 

2-9 

0-241 

278 

0-161 

Tin, .... 


456 

3-16 

0-263 

358 

2-48 

0-207 

239 

0-138 

Lead, - - - - 


710 

4-92 

0-41 

557 

3-87 

0-322 

371 

0-215 

Zinc, ... - 


440 

3-05 

0-254 

345 

2-4 

0-2 

230 

0-133 

Copper, • - - 


556 

3-85 

0-321 

436 

3-03 

0-252 

291 

0-168 

Mercury,. - - 


850 

5-9 

0-491 

666 

4-63 

0-385 

445 

0-257 

Stone, common, 


156 

1-08 

0-09 

122 

0-85 

0-071 

82 

0-047 

Clay, - • - - 


135 

0-936 

0-078 

106 

0-735 

0-061 

70 

0-04 

Earth, compact, 


127 

0-88 

0-0733 

99 

0-692 

0-058 

66 

0-038 

Earth, loose, - 


95 

0-66 

0-055 

74 

0-517 

0-043 

50 

0-029 

Oak, dry, - - 


58 

0-4 

0-033 

44 

0-316 

0-026 

30 

0-017 

Pine, - - - - 


30 

0-208 

0-017 

24 

0-163 

0-014 

16 

0-009 

Mahogany, - - 


66 

0-457 

0-038 

52 

0-36 

0-03 

34 

0-02 

Coal, stone, - - 


54 

0-375 

0-031 

42 

0-294 

0-024 

28-2 

0-016 

Charcoal,- - - 


27-5 

0-19 

0-016 

21 

0-15 

0-012 

14-4 

0-008 


To Find the Weight and Capacity by this Table 

RULE. The product of the dimensions in feet or in inches, as noted in the 
columns, multiplied by the tabular coefficient, is the capacity of the solid, or 
weight in pounds avoirdupois. 

Example 1. A cistern is 6 feet long, 27 inches wide, and 20 inches deep. 
How many gallons of liquid can he contained in it ? 


6X27 X20X0-052 = 168 48 gallons. 

Example 2. A cast-iron cylinder is 4-5 feet long, and 7 "5 inches diametei 
Required the weight of it ? 

4-5X7-5 ,, X2-45 = 620 pounds. 


SELECTION OF 

Blue. Real Ultramarine. 

“ French Blue. 

“ Indigo. 

“ Cobalt Blue. 

Green. Olive Green. 

Yellow. Cadmium. 

“ Gamboge. 

“ Ochre. 

Red. Carmine. 

“ Crimson Lake. 


WATER COLOURS. 

Red. Rose Madder. 

“ Light Red. 

Brown. Vandyke. 

“ Brown Madder. 

Black. India Ink. 

“ Blue Black. 

“ Ivory Black. 

“ Lamp Black. 

White. Chinese White. 











































352 


Paper, Tin and Grass. 


13X16 

inches. 

Columbier, 

34 X 23 inches. 

20 “ 15 

66 

Atlas, . 

33 “ 26 

66 

22 “ 17 

u 

Theorem, 

34 “ 28 

66 

24 “ 19 

u 

Double Elephant, 

40 “ 26 

u 

27 “ 19 

66 

Antiquarian, . 

52 “ 31 

u 

30 “ 21 

u 

Emperor, . 

40 “ 60 

66 

28 “ 22 

u 

Uncle Sam, 

48 “ 120 

46 


PAPER. 

1 ream = 20 quires = 480 sheets. 

1 quire = 24 sheets. 

Drawing Paper. 

Csp, • « , 

Demy, 

Medium, 

Royal, 

Super Royal, . 

Imperial, . 

Elephaut, 

Continuous Colossal Drawing Paper , No. A and No. B, 56 inches wide, and of any 
required length. No A of this paper is excellent for mechanical drawings. Price, 
from 40 to 50 cents per yard. 

Tracing Paper. 

Double Crown,.30 by 20 inches.'! Glazed or CrystaI< 

Double Double Crown, . . . 40 “ 30 f Yellow or Blue Wove. 

Double Double Double Crown, . . 60 “ 40 “ J xenowor mue wove. 

Finest French Vegetable Tracing Paper. 

Grand Raisin (or Royal), 24 in. by 18. Grand Aigle, 40 in. by 27. 

Mounted Tracing Paper. 

This paper is mounted on cloth, and is still transparent; it will take ink and 
water-colors. It is 38 inches wide, and of any required length. 

Vellum Writing Cloth. 

Adapted for every description of tracing; it is transparent, durable and strong. 
It is 18 to 38 inches wide, and of any required length. 

'Weight and Marks of English Tin-plates. 




Plates 

Length 

Weight 


Plates 

Length 

Weight 

Brand. 

per 

and 

per 

Brand. 

per 

and 

per 



Box. 

Breadth. 

Box. 


Box. 

Breadth. 

Box. 



No. 

In. 

Lbs. 


No. 

In. 

Lbs. 

1 C. 

• 

225 

131X10 

112 

1 XX. . . 

225 

13|X10 

161 

2 0 . . 

• 

225 

13y “ 91 

105 

1 XXX. . 

225 

13$ “ 10 

182 

3 C. 


225 

121 “ 91 

98 

1 xxxx. . 

225 

13$ “ 10 

203 

HC. . 

e 

225 

131 “ 10 

119 

i xxxxx. 

225 

13$ “ 10 

224 

H X. 

• 

225 

131“ 10 

157 

1 xxxxxx. 

225 

13$ “ 10 

245 

IX. . 

# 

225 

131 “10 

140 

DC. 

100 

16$ “ 12 i 

98 

2 X. 

• 

225 

131“ 91 

133 

DX. . 

100 

16$ “ 12 i 

126 

3 X. . 

# 

225 

121 “ 91 

126 

DXX. . 

100 

16$ “ 12 ^ 

147 

Leaded IC. 

112 

20 “14 

112 

DXXX. . 

100 

16$ “ 12 J- 

168 

66 

IX. 

112 

20 “14 

140 

DXXXX. 

100 

16$ “ 12 A 

189 

ICW. 

# 

225 

131 “ 10 

112 

SDC. 

200 

15 “11 

168 

IXW. 

• 

225 

131 “ 10 

140 

SDX. . 

200 

15 “11 

188 

CSDW. 


200 

15 “11 

168 

SDXX. . 

200 

15 “11 

- 209 

CMW. 

• 

100 

161“ 121 

105 

SDXXX. 

200 

15 “11 

230 

XIIW. 

# 

100 

161 “ 121 

126 

S DXXXX. 

200 

15 “11 

251 

TT. . 

# 

450 

131 “ 10 

112 

SDXXXXX. 

200 

15 “11 

272 

XTT. 

• 

450 

131 “ 10 

126 

SDXXXXXX. 

200 

15 “11 

293 


When the plates are 14 by 20 inches, there are 112 in a box. 

Thickness and Weight of Window Glass. 


Number of the glass or weight in ounces per square foot. 


12 1 

13 1 

15 1 

16 1 

17 1 

19 1 

21 1 

24 1 

1 26 1 

32 1 

.059 | 

.063 | 

.071 1 

.077 | 

.083 | 

| .091 [ 

.loo | 

.111 | 

.125 I 

l .154 1 


36 

.167 


42 

.200 


Thickness iu decimals of an inch. 























The Atmosphere. 


353 

—i 


THE ATMOSPHERE. 

The mean height of the atmosphere is about 302 feet greater at the equator than 
j at the poles, which is caused by the difference of the earth’s attraction at the two 
j places, and also by centrifugal force. 

1 The mean height of the atmosphere in 45° latitude is 60158.6 feet; at the poles, 
i60 :07.6 feet, and at the equator, 60309.6 feet. 

The temperature of the atmosphere is greatest at the surface of the earth, and 
'\ decreases with the height above the surface. The compression of the air by the 
: upper layers of the atmosphere generates heat in the lower layers, as explained in 
| the article on Air and Heat. The rays of light from the sun, passing through a 
j denser air near the surface of the earth, also generate more heat by friction, as it 
| were. The temperature of congelation of water being 32°, which is marked by the 
' perpetual snow-line on high mountains, as shown in the accompanying table. 

Heights of Snow-Line in Different Latitudes. 


i 


Latitudes of snow-line on high mountains. 


1 16 ° 1 

25° 1 

35° 

40° 

| 45° 1 

1 55° 

1 65° 

75° 

| 14,760 | 

12,560 | 

10,290 | 

9,000 

| 7,670 

j 5,030 

| 2,230 

| 1,016 


85° 

120 


Heights of snow-line in feet above the sea. 


New-fallen snow occupies eight times its volume in water. 

Heat is constantly absorbed from the atmosphere by evaporation of water on the 
surface of the seas, which heat is carried up and warms the atmosphere above ; heat 
is also absorbed by support of the growth of vegetation on land. It is this opera¬ 
tion of consuming and generating heat which causes the wiuds and difference of 
weather. 

As the atmosphere is a material substance, it is subject to the action of the force 
of gravity, which causes a pressure of 14.75 pounds to the square inch at the level 
of the sea; or a column of air one square inch base and of the height of the atmo¬ 
sphere weighs 14.75 pounds, which balances an equal weight of a column of mer¬ 
cury 30 inches high at tiie temperature of 60° Fahr., or a column of water of 34 
feet high. 

Columns of Air, Mercury and Water. 


A is a vessel full of mercury, in which is placed verti¬ 
cally a glass tube about 3 feet high above the surface Z; 
in the glass tube is fitted an air-tight piston a, just one 
square inch area, which can be moved by the piston-rod c; 
now the piston stand is at a on the level Z, and in contact 
with the mercury in the tube ; raise the piston by the pis¬ 
ton-rod and handle c, the mercury in the tube will follow 
until the heig-ht of 30 inches, the piston still continues to 
move higher in the tube, but the mercury will maintain its 
position at 30 iuches from l. Now it may be supposed that 
it is some force of the piston that draws the mercury up in 
the tube; if so, why did it separate at 30 inches? If the 
column becomes too heavy, it could separate at Z, and the 30 
inches, of mercury follow the piston ; as this is not the case, 
but the weight of the atmosphere pressing on the surface Z 
and forcing the mercury up in the tube until they (the 
mercury and the atmosphere) come in equilibrium, which 
occurs at the 30 inches; and the piston only served to 
remove the atmospheric pressure in the tube; hence we 
have the weight of a column of atmospheric air with one 
square inch base equal to the weight of a column of mer¬ 
cury 30 inches high and one sq. in. bjise. One cubic inch 
of mercury at 60° Fahr. weighs 0.941 pounds; this, multi- 
i plied b,v the height, 30 inches, gives 14.73 pounds, the weight of the column of 
i mercury or atmosphere; this is generally termed “ the atmospheric pressure per 
square inch.*’ 

The specific gravity of mercury at .60° Fulir. is 13.58, and 

= 33.95 feet, 

12 

•the height of a column of water required to balance the atmosphere. 
























354 


Wind, Aerodynamic. 


WIND, AERODYNAMIC. 

The motions and effects of gases by the force of gravity are precisely the same 
as that of liquids. (See Hydraulics.) 

The altitude or head of the atmosphere at uniform density will be the altitude of 
a column of water 33.95 feet, divided by the specific gravity of the air, 0.0012046, or, 

33.95 00100 e A 

-- = 28133 feet. 

0.0012046 

The velocity due at the foot of this head will be — 

V = 8.02 j/28183 = 1346.4 feet per second, the velocity at which the air will pass 
into a vacuum. 

Velocity of Wind. 

When air passes into an air of less density, the velocity of its passage is meas¬ 
ured by the difference of their density. 

H and h = density of the air in inches of mercury; t — temperature at the time 
of passage ; and V = velocity of the wind in feet per second. 


V= 1346.4 


II — h 
h 


(l + 0.00208*), . 


6 . 


The force of wind increases as the square of its velocity. 

a = area exposed at right angles to the wind in square feet; F= force of the 
wind in pounds; H = horse-power, and v = velocity of the plane a in direction 
of the wind, -f- when it moves opposite, and — when it moves with the wind. 

7. 
9. 


F= 0.002288a V 2 , 

F — 0.002288a( V zb v) 2 , 8. 


when v = o, 

, r &v( V =t v) 2 

l1 = 


241400 


Example. A rail-train running ENE 25 miles per hour exposes a surface of 
1000 square feet to a pleasant brisk gale NE by E. Required the resistance to the 
train in the direction it moves, and the horse-power lost. 

ENE — N E bv N = 3 points = 33° 45'; V — 14 feet per second, a brisk gale ; 
v = 25 X 1.467 = 36.6 feet per second, and F = 0.002288 sin. 233° 46' X 1000 (14 + 
cos. a3° 45' X 36.6)2 = 305.1 pounds. 


H = 


305.1 X 36.6 
5'50 


= 20 horses. 


Table of Velocity and Force of Wind, in Pounds per Square 

Inch. 


Miles 

Feet 

Force 

per 

Common Application of 

Miles 

Feet 

Force 

per 

per 

hour. 

second 

sq. ft. 
pound. 


the force of Wind. 

hour. 

second 

sq. ft. 
pound. 

i 

1.47 

0.005 

f Hardly percept- 

18 

26.4 

1.55 



ible. 

20 

29 34 

1.968 

2 

3 

2.93 

4.4 

0.020 

0.044 

i 

< 

-Just perceptible. 

25 

30 

36.67 
44 01 

3.075 

4429 

4 

5.87 

0.079 


35 

51.34 

6 027 

5 

7.33 

0 123 


, Gentle pleasant 

4') 

58.68 

7.873 

6 

8.8 

0.177 

• 

wind. 

45 

66.01 

9.963 

7 

10.25 

0.241 



50 

73.35 

12.30 

8 

11.75 

0.315 



55 

80.7 

14.9 

9 

13.2 

0 400 



60 

88.02 

17.71 

10 

12 

14.67 

17.6 

0.492 

0.708 


Pleasant brisk 
gale. 

66 

70 

95.4 

102.5 

20.85 

24.1 

14 

20.5 

0 964 


75 

no 

27.7 

15 

22.00 

1.107 



80 

117.36 

31.49 

16 

23.45 

1.25 



100 

140.66 

50. 


Common Application of 
the force of Wind. 


Yery brisk. 
High wind. 


Very high. 
Storm. 


Great storm, 
j- Hurricane. 
Tornado. 




















The Barometer. 


355 


THE BAROMETER. 

The bai’ometer measures the pressure of the atmosphere, as described in the 
former editions of this Pocket hook. 

The English have graduated the barometer to indicate weather as follows : 


Barometer in inches. 
At 28.3 = 

At 28.7 = 

At 29.1 = 

At 29.5 = 

At 29.9 = 

At 30.3 = 

At 30.7 = 


Weather. 

Stormy. 

Much rain. 

Rain. 

Change of weather. 
Fair weather. 

Set fair. 

Very dry. 





The following guides in predicting weather-changes are selected from 
the “ Barometer Manual” of the London Board of Trade: 

I. If tiie mercury, standing at thirty inches, rise gradually while the 
thermometer falls, and dampness becomes less, N.W., N. or N.E. wind ; less 
wind or less snow and rain may be expected. 

II. If a fall take place with a rising thermometer and increasing damp¬ 
ness, wind and rain may be expected from S.E., S. or S.W. A fall in 
winter with a low thermometer foretells snow. 

III. An impending north wind, before which the barometer often rises, 
may be accompanied with rain, hail or snow, and so forms an apparent 
exception to the above rules, for the barometer always rises with a north 
wind. 

IV. The barometer being at 29| inches, a rise foretells less wind or 
a change of it northward, or less wet. But if at 29 inches, a fast first rise 
precedes strong winds or squalls from N.W., N. or N.E., after which a gradual rise 
with falling thermometer, a S. or S.W. wind will follow, especially if the rise of 
the barometer has been sudden. 

V. A rapid barometric rise indicates unsettled, and a rapid fall stormy, weather 
with rain or snow; while a steady barometer, with dryness, indicates continued 
fine weather. 

AT. The greatest barometric depressions indicate gales from S.E., S. or S.W.; the 
greatest elevations foretell wind from N.W., N. or N.E., or calm weather. 

ATI. A sudden fall of the barometer, with a westerly wind, is sometimes followed 
with a violent storm from the N.W., N. or N.E. 

VIII; If the wind veer to the south during a gale from the E. to S E., the barom¬ 
eter will continue to fall until the wind is near a marked change, when a lull may 
occur. The gale may afterward be renewed, perhaps suddenly and violently ; and 
if the wind then veer to the N.AV., N. or N.K., the barometer w ill rise and the ther¬ 
mometer fall. 

IX. The maximum height of the barometer occurs during a north-east wind, and 
the minimum during one from the south-west; hence these points may be consid¬ 
ered the poles of the wind. The range between these two heights depends on the 
direction of the wind, which causes, on an average, a change of half an inch; on 
the moisture of the air, which produces, in extreme cases, a change of half an inch ; 
and on the strength of the wind, which may influence Ihe barometer to the extent 
of two inches. These causes, separately or conjointly with the temperature, pro¬ 
duce either steady or rapid barometric variations, according to their force. 


















i 9$ 


Hygrometry. 


HYGROMETRY. 

On the Humidity and other Properties of Air , deduced from Glaisher's Tables 
of the Greenwich Observatory. 

Mason’s hygrometer, consisting of wet and dry bulb thermometers, is considered 
the best for determining humidity in the air and the dew-point. 

Example. The temperature of the air being 75°, and the wet-bulb thermometer 
showing 63°, or 12° cold; barometer 30 inches. Required, the humidity of the 
air, the dew-point, weight of vapor per cubic foot, and the weight of a cubic foot 
of the air in grains troy ? 

Table I., 75° and 12° cold = 55 per cent, of humidity. 

Table II., “ “ = 57° temperature of dew-point. 

Table III., weight of dry air = 516.7 grains per cubic foot. 

“ “ “ saturated air = 511.4 “ “ “ 

Difference = 5.3 X 0-55 = 2.915 grains. 

Weight of the air 511.4 2.9 = 514.3 grains per cubic foot. 

Table III., 9.31 + 0.55 == 5.12 grains of vapor per cubic foot. 

The weight of air of equal temperature and humidity is inverse as the height 
of the barometer. 

TABLE I. 


Humidify of the Air, or Percentage of Pull Saturation, 

At Different Temperatures , indicated by the Dry and Wet Bidbs 
of the Hygrometer ( Glaisher). 


Temp, of 
the air, 

Difference 

in 

Temperature, or Cold on the Wet 

-bulb Thermometer. 

Fahr. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

30 

86 

73 

55 





















35 

91 

83 

76 

70 

64 

57 

53 

48 
















40 

93 

86 

79 

74 

68 

63 

53 

53 

50 

46 














45 

93 

86 

80 

74 

69 

64 

59 

55 

51 

48 














50 

93 

87 

81 

76 

71 

66 

61 

57 

53 

49 

46 













55 

94 

88 

83 

78 

75 

69 

65 

60 

56 

53 

50 

49 

46 

44 

41 

39 

36 







60 

94 

89 

84 

80 

75 

71 

67 

63 

59 

56 

53 

50 

17 

45 

42 

40 

38 

35 

33 





65 

95 

89 

85 

81 

76 

72 

69 

65 

61 

5S 

55 

52 

49 

47 

44 

42 

40 

37 

35 

34 

32 

30 

28 

70 

95 

91 

86 

82 

78 

74 

71 

67 

64 

61 

58 

55 

52 

49 

47 

45 

42 

40 

37 

35 

34 

31 

29 

75 

95 

90 

86 

82 

78 

74 

71 

68 

64 

61 

58 

55 

52 

49 

48 

47 

44 

41 

39 

37 

35 

32 

30 

80 

95 

90 

87 

83 

79 

75 

72 

68 

65 

62 

59 

56 

53 

50 

49 

48 

44 

42 

40 

38 

36 

33 

31 

85 

96 

91 

87 

83 

79 

75 

72 

68 

65 

62 

59 

56 

54 

51 

49 

46 

44 

42 

40 

38 

36 

34 

32 

90 

96 

91 

87 

83 

79 

75 

72 

68 

65 

62 

59 

56 

54 

.51 

49 

46 

44 

42 

40 

38 

36 

34 

32 


Percentage of Humidity. 


TABLE II. 

Temperature of tine Dew-point, 

At Different States of the Hygrometer. 


Temp, of 


Difference in Temperature, or Cold on the Wet-bulb Thermometer. 


Fahr. 

i 

2 

3 

4 

5 

6 

7 

8 9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 ?3 

30 

25 

21 





















35 

32 

30 

27 

25 

22 

20 

17 

15 13 















40 

37 

35 

33 

31 

29 

27 

25 

22 20 18 














45 

43 

41 

39 

37 

34 

32 

30 

28 26 24 22 













50 

48 

46 

44 42 

40 

38 

36 

34,32 30 28 

26 












55 

53 

52 

50 48 46 

45 

43 

41 40 38 36 

34 

33 

31 

29 

28 

26 

24 






60 

58 

56 

55 

53 

51 50 

48 

46! 15 43 41 

39:38 

36 

34 

33 

31 

29 

28 

26 




65 

63 

62 

60 

58 

57 55 

54 

52'50 49 47 46 44 

43 

41 

39 

38 

36 

34 

33 

31 

30 

28 

70 

68 

67 

65 

64 

62 

61 

59 

58 56 55 53 52 

50 

49 

47 

46 

44 

43 

41 

40 

38 

37 

35 

75 

73 

72 

70 

69 

67 

66 

64 

63 61 60 

58,57 

55 

54 52 

51 

49 

48 

1C 

45 

43 

42 

40 

80 

78 

77 

75 

74 

72 

71 

69 

68 66 65 

63 62 60 59 57 

56 

54 

53 

51 50148 

47 45 

85 

84 

83 82 

81 

80 

79 

78 

77 76 75 74 73 72 

71 70 

69 

68167 

66 | 65 64 

03 62 

90 

89 

88.87 

86 

85 

84 

83 

82 81 

80,79 78 77 AG.75 

74 

73,72 

71,70,69 

(58167 


Temperature of Dew-point. 











































































Hygkometry. 


a<57 


TABLE III. 


Properties of Air, by GlaisUer, Greenwich Observatory. 

Barometer 30 inches , at 60° Fahrenheit. 


Temp. 

Force of 

Weight 
of vapor 

Wt. per cub. ft. 

Temp. 

Force ot 

Weight 
of vapor 

Wt. per cub.ft. 

of the 
air. 

v<ipoi in 
inches of 

per cub. 
ft. of 

Dry 

Satu¬ 

rated 

of tiie 
air. 

vapor in 
inchesoi 

per cub. 
foot of 

Dry 

Sat’d 

mercurjr 

sat. air. 

air. 

air. 

mercury 

sat. air. 

air. 

air. 

Fahr. 

Inches. 

Grains. 

Grains. 

Grains. 

Fahr. 

Inches. 

Grains. 

Grains. 

Grains. 

10 c 

0.089 

1.11 

590.0 

589.4 

52 ° 

0.400 

4.56 

540.5 

537.9 

11 

0.093 

1.15 

588.7 

588.1 

53 

0.414 

4.71 

539.4 

536.7 

12 

0.096 

1.19 

587.5 

586.8 

54 

0.428 

4.86 

538.3 

535.5 

13 

0.100 

1.24 

586.2 

585.5 

55 

0.442 

5.02 

537.3 

534.4 

14 

0.104 

1.28 

584.9 

584.2 

56 

0.458 

5.18 

536.2 

533.2 

15 

0.108 

1.32 

583.7 

582.9 

57 

0.473 

5.34 

535.1 

532.1 

16 

0.112 

1.37 

582.4 

581.6 

58 

0.489 

5.51 

534.1 

530.9 

17 

0.116 

1.41 

581.1 

5S0.3 

59 

0.506 

5.69 

533.0 

529.8 

18 

0.120 

1.47 

579.9 

579.1 

60 

0.523 

' 5.87 

532.0 

528.6 

19 

0.125 

1.52 

578.7 

577.8 

61 

0.541 

6.06 

530.9 

527.5 

20 

0.129 

1.58 

577.4 

576.5 

62 

0.559 

6.25 

529.9 

526.3 

21 

0.134 

1.63 

576.2 

575.3 

63 

0.578 

5.45 

528.8 

525.2 

22 

0.139 

1.69 

575.0 

574.0 

64 

0.597 

6.65 

527.8 

524.0 

23 

0.144 

1.75 

573.7 

572.7 

65 

0.617 

6.87 

526.9 

522.9 

24 

0.150 

1.81 

572.5 

571.5 

66 

0 . 63“8 

7.08 

525.8 

521.7 

25 

0 . 155 - 

1.87 

571.3 

570.2 

67 

0.659 

7.30 

524.7 

520.6 

26 

0.161 

1.93 

570.1 

569.0 

68 

0.681 

7.53 

523.7 

519.4 

27 

0.167 

2.00 

568.9 

567.7 

69 

0.704 

7.76 

522.7 

518.3 

28 

0.173 

2.07 

567.7 

566.5 

70 

0.727 

8.00 

521.7 

517.2 

29 

0.179 

2.14 

566.5 

565.3 

71 

0.751 

8.25 

520.7 

516.0 

30 

0.186 

2.21 

565.3 

564.1 

72 

0.776 

8.50 

519.7 

514.9 

31 

0.192 

2.29 

564.2 

562.8 

73 

0.801 

8.76 

518.7 

513.7 

32 

0.199 

2.37 

563.0 

561.6 

74 

0.827 

9.04 

517.7 

512.6 

33 

0.207 

2.45 

561.8 

566.4 

75 

0.854 

9.31 

516.7 

511.4 

34 

0.214 

2.53 

560.7 

559.2 

76 

0.882 

9.60 

515.7 

510.3 

35 

0.222 

2.62 

559.5 

558.0 

77 

0.910 

9.89 

514.7 

509.2 

36 

0.230 

2.71 

558.3 

556.8 

78 

0.940 

10.19 

513.8 

508.0 

37 

0.238 

2.80 

557.2 

555.6 

79 

0.970 

10.50 

512.8 

506.9 

38 

0.246 

2.89 

556.0 

554.4 

80 

1.001 

10.81 

511.8 

505.7 

39 

0.255 

2.99 

554.9 

553.2 

81 

1.034 

11.14 

510.9 

504.6 

40 

0.264 

3.09 

553.8 

552.0 

82 

1.067 

11.47 

509.9 

503.4 

41 

0.274 

3.19 

552.6 

550.8 

83 

1.101 

11.82 

508.9 

502.3 

42 

0.283 

3.30 

551.5 

549.6 

84 

1.136 

12.17 

508.0 

501.1 

43 

0.293 

3.41 

550.4 

548.4 

85 

1.171 

12.53 

507.0 

500.0 

44 

0.304 

3.52 

549.3 

547.2 

86 

1.209 

12.91 

506.1 

498.9 

45 

0.315 

3.64 

548.1 

546.1 

87 

1.247 

13.29 

505.1 

497.7 

46 

0.326 

3.76 

547.0 

544.9 

88 

1.286 

13.68 

504.2 

496.6 

47 

0.337 

3.88 

546.0 

543.7 

89 

1.326 

14.08 

503.2 

495.4 

48 

0.349 

4.01 

544.8 

542.5 

90 

1.368 

14.50 

502.3 

494.3 

49 

0.361 

4.14 

543.7 

541.3 

91 

1.411 

14.91 

501.3 

493.2 

50 

0.373 

4.28 

542.6 

540.2 

92 

1.456 

15.33 

500.4 

492.0 

51 

0.386 

4.42 

541.5 

539.0 

93 

1.502 

15.76 

499.4 

491.9 



























358 


Climate and Seasons, 


MEAN TEMPERATURE AT DIFFERENT SEA¬ 
SONS OF THE YEAR. 

* 


Locations. 

Year. 

Lean Te 

Spring. 

mperat 

Sum. 

ure, Fa 

Autm. 

lr. 

Wmt'r. 

Hemis¬ 

phere. 

Height 
ab. sea. 
Feet. 

Algiers,. 

63.0 

63.0 

74.5 

70.5 

54.0 

N. 

310 

Berlin, . . . . • 

47.5 

46.4 

63.1 

47.8 

30.6 

N. 

128 

Berne,. 

46.0 

45.8 

60.4 

47.3 

30.4 

N. 

1918 

Boston,. 

49 

48 

66 

53 

28 

N. 

71 

Buenos Ayres, 

62.5 

59.4 

73.0 

64.6 

52.5 

S. 


Cairo,. 

72.3 

71.6 

84 6 

74.3 

58.5 

N. 


Calcutta,. 

78.4 

82.6 

83.3 

80.0 

67.8 

N. 


Canton, . 

69.8 

69.8 

' 82.0 

72.9 

54.8 

N. 

io 

Christiania, .... 

41.7 

39.2 

59.5 

42.4 « 

25.2 

N. 

74 

Cape of Good Hope, 

66.4 

63.5 

74.1 

66.9 

58.6 

S. 


Constantinople, 

56.7 

51.8 

73.4 

60.4 

40.6 

N. 

150 

Copenhagen, .... 

46.8 

43.7 

63.0 

48.7 

31.3 

N. 

20 

Edinburgh, .... 

47.5 

45.7 

57.9 

48 0 

38.5 

N. 

288 

Jerusalem, .... 

62.2 

60.6 

72.6 

66.3 

49.6 

N. 

2500 

Jamaica (Kingston), 

79.0 

78.3 

81.3 

80 0 

76.3 

N. 

10 

Lima, Peru, .... 

66.2 

63.0 

73.2 

69.6 

59.0 

S. 

511 

Lisbon. 

61.5 

599 

71.1 

62.6 

52.3 

N. 

236 

London, .... 

50.7 

49 1 

62.8 

51.3 

39.6 

N. 

50 

Madeira (Funchal), . 

65.7 

63 5 

70.0 

67.6 

61.3 

N. 


Madrid, . ... “ 

57.6 

57.6 

74.1 

56.7 

42.1 

N. 

2175 

Mexico, City, .... 

60.5 

53.6 

63.4 

65.2 

60.1 

N. 

6990 

Montreal, .... 

43.7 

44.2 

69.1 

47.1 

17.5 

N. 


Moscow,. 

38.5 

43.3 

62.6 

34.9 

13.5 

N. 

480 

Naples. 

61.5 

59.4 

74.8 

62.2 

49.6 

N. 

180 

New Orleans, .... 

72 

73 

84 

72 

58 

N. 

20 

New York, .... 

53 

50 

72 

56 

33 

N. 

20 

New Zealand, .... 

59.6 

60.1 

66.7 

58.0 

53.5 

S. 


Nice. 

60.1 

55.9 

72.5 

63.0 

48.7 

N. 


Nicolaief (Russia), . 

48.7 

49.3 

71.2 

50.0 

25.9 

N. 


Paramatta (Australia), 

64.6 

66.6 

73 9 

64.8 

54.5 

S. 


Palermo,. 

63.0 

59.0 

74.3 

66.2 

52.5 

N. 

180 

Pekin, China, 

626 

56.6 

77.8 

54.9 

29 

N. 

97 

Paris,. 

51.4 

50.5 

64.6 

52.2 

37.9 

N. 

210 

Philadelphia, 

55 

52 

76 

57 

34 

N. 

30 

Quito, Ecuador, 

60.1 

60.3 

601 

62.5 

59.7 

S. 

9560 

Rio Janeiro, .... 

73.6 

72.5 

79.0 

74.5 

68.5 

S. 

10 

Rome,. 

59.7 

57.4 

73.2 

61.7 

46.6 

N. 

174 

San Francisco, 

57.6 

58 

59 

60 

53 

N. 

150 

St. Petersburg,.... 

38.3 

35.1 

60.3 

40.5 

16.7 

N. 

10 

Stockholm, . . . 

42.1 

38 3 

61.0 

43.7 

25.5 

N. 

134 

Trieste,. 

55.8 

53.8 

71.5 

56.7 

39.4 

N. 

288 

Turin, .... 

53.1 

53.1 

71.6 

53.8 

33.4 

N. 

915 

Vienna. 

50.7 

49.1 

62.8 

51.3 

39.6 

N. 

480 

Warsaw, . . 

45.5 

41.6 

63.5 

46.4 

27.5 

N. 

397 

Washington, .... 

59 

60 

79 

58 

38 

N. 

. . . 


Seasons. 


Southern Latitude. 

Seasons. 

Northern Latitude. 

December, 
Mm cli, 
June, 

September, 

J anuary, 
April. 
July, 
October, 

February, 

May, 

August, 

November, 

Summer. 

Fall. 

Winter. 

Spring. 

June, 

September, 

December, 

March, 

July, 

October, 

January, 

April, 

August. 

November. 

February. 

May. 















































Rain and Melted Snow. 


359 


Rain and Melted Snow. 


Fall in Inches at Different Places. 


Locations. 

Year. 

Spring. 

Summ’r. 

Fall, 

Winter, 

Albany, North America, 

40.67 

9.79 

12.3 

10.3 

8.30 

Algiers,. 

37.01 

8.34 

0.60 

10.3 

17.8 

Baltimore, North America, 

42.00 

11.2 

11.1 

10.52 

9.31 

Berlin, Prussia, 

23.56 

5.66 

7.21 

5.45 

5.24 

Bergen, Norway, .... 

87.61 

15.7 

18.6 

29.8 

23.5 

Bombay, India. 

110 . 





Boston, North America, 

44.48 

10.8 

11.8 

12.57 

9.89 

Buffalo, “ 

27.35 

5.90 

8.45 

7.48 

5.52 

Canton, China, .... 

69.3 

18.8 

27.9 

19.3 

3.3 

Charleston, North America, . 

48.29 

8.60 

18.7 

11.6 

9.40 

Copenhagen, .... 

18.35 

2.S4 

6.86 

5.13 

3.52 

Dover, Englaud, .... 

38. 

• • • 

• • • 

• * e 

• • 

Dublin, Ireland, 

25. 

, , 

• * # • 



Edinburgh, Scotland, . . . 

28. 

• • • 

• • • 

O • • 

• • • 

England,. 

33. 

• • • 

* • • 

• • • 

• • • 

Glasgow, . 

28.9 

5.43 

7.13 

8.95 

7.39 

Granada (Colombia), 

115. 

• • • 

• • • 

• • • 

• • • 

Liverpool, ..... 

34.1 

6.19 

9.78 

10.8 

7.32 

Lima, Peru, .... 

13.5 

5.1 

0.2 

1.2 

7.0 

London, . 

20.69 

4.09 

6.00 

6.15 

4.45 

Madeira Islands, 

30.87 

5.11 

2.30 

6.96 

16.5 

Manchester, England, 

36. 

7. 

9. 

11 . 

9. 

Milano, Italy, .... 

38. 

9.04 

9.18 

11.7 

8.05 

Mississippi State, .... 

53.00 

10.9 

14.2 

9.50 

18.4 

New York, .... 

42.23 

11.5 

11.3 

10.3 

9.63 

New Orleans, .... 

52.31 

13.3 

16.1 

10.8 

12.6 

Ohio, State, .... 

39.69 

10.4 

10.9 

9.03 

6.91 

Pekin, China, .... 

26.9 

2.67 

20.5 

3.22 

0.53 

Peru (Interior),Carabaya, 

355. 

88 . 

120 . 

87. 

60. 

St. Petersburg, .... 

17.65 

2.89 

6.73 

5.11 

2.93 

Paris,. 

22.64 

5.53 

5.92 

6.51 

4.68 

Philadelphia, .... 

48.00 

13. 

12 . 

11 . 

12 . 

Rio Janeiro, Brazil, . 


. . 

• • • 

• • • 

10.76 

Rome, Italy, .... 

30.87 

7.27 

3.4 

10.9 

9.3 

Stockholm, .... 

19.67 

2.17 

7.81 

6.94 

2.75 

Tiflis, Caucasus, .... 

19.26 

6.25 

7.62 

3.51 

1.88 

Washington, .... 

41.20 

10.4 

10.5 

10.2 

11.1 

San Francisco, California, . 

83. 

22 . 

1 . 

15. 

45. 


Volume of Evaporation and Rain-Fall. 

Inches X 2,323.200 = cubic feet per square mile. 
Inches X 17,335,019 = gallons per square mile. 
Inches X 3630 = cubic feet per acre. 


Length in Miles of the Principal Rivers. 


Europe. 


North and South 
America. 


Asia and Africa. 


Volga, Russia, . . 

2000 

Missouri, . . . 

2900 

Yang-tse-kiang, . 

2800 

Danube, .... 

1600 

Mississippi, . . 

2800 

Lena, .... 

2600 

Don and Dnieper, . 

1000 

Mackenzie’s, . . 

2500 

Obe, Hoangho, . 

2500 

Rhine, .... 

950 

St. Lawrence, . 

2200 

Yeuesei, . . . 

2300 

Dwina,. 

700 

Rio Grande, . . 

1800 

Amor, .... 
Cambodia, . . . 

2200 

Petchora, Elbe. Loire, 

600 

Colorado, Cal., . 

1100 

2000 

Vistula. Tagus, . . 

550 

Alabama, . . . 

600 

Indus, Irrawaddy, 

1700 

Dniester, Guadiana, 

500 

Amazon, . . . 

3600 

Nile,. 

3000 

Rhone, Po, Seine, . 

450 

Rio de la Plata, . 

2250 

Niger or Joliba, 

2600 

Mezene, Desna, . 

400 

Orinoco, . . . 

1500 

Senegal, . . . 

1200 

Dahl, Bug, .... 

300 

Araguay, . . . 

1100 

Orange, . . . 

1000 

Thames, .... 

233 

Magdalena, . . 

900 

Gambia, . . . 

700 


































®so 


ETAFoSATjam 


Evaporation oil tlie Surface of Water in tlie Open Air. 

When the surface of water is freely exposed under the atmosphere, the dry air in 
contact with it becomes charged with vapor, and consequently becomes lighter 
(see Table, page 357), rises, and gives place to drier air, to repeat the same opera¬ 
tion, by which moisture is constantly carried up into the air from the surface of 
the water. The rate of this evaporation depends upon the temperature of the 
water, the dryness, the temperature and the velocity of the air. 

Evaporation of Water in Decimals of an Xncli, per 24 Hours, 

on the surface of fresh-water lakes, rivers and canals, at different temperatures of 
the water and currents of the air. 


Water. 


Telocity of wind in miles per hour on the water. 


Temp. 

Calm. 

10 

20 

30 

40 

50 

60 

32° 

0.012 

0.014 

0:016 

0.017 

0.019 

0.021 

0.023 

35 

0.020 

0.023 

0.026 

0.029 

0.032 

0.035 

0.038 

40 

0.040 

0.046 

0.052 

0.058 

0.064 

0.070 

0.076 

45 

0.068 

0.078 

0.088 

0.098 

0.109 

0.119 

0.129 

50 

0.100 

0.115 

0.130 

0.145 

0.160 

0.175 

0.190 

55 

0.133 

0.153 

0.173 

0.193 

0 213 

0.233 

0.253 

60 

0.177 

0.203 

0.230 

0.256 • 

0.283 

0.310 

0.336 

65 

0.225 

0.259 

0.292 

0.320 

0.360 

0.394 

0.427 

70 

0.278 

0.320 

0.361 

0.404 

0.444 

0.486 

0.527 

75 

0.335 

0.385 

0.435 

0.485 

0.535 

0.585 

0.635 

80 

0.400 

0.460 

0.520 

0.580 

0.640 

0.700 

0.760 

85 

0.468 

0 538 

0.608 

0.679 

0.749 

0.819 

0.8S9 

90 

0.540 

0.621 

0.703 

0.784 

0.865 

0 946 

1.025 

95 

0.620 

0.713 

0.808 

0,900 

0.995 

1.088 

1.180 

100 

0.700 

0.805 

0.912 

1.015 

1.123 

1.225 

1.332 


The evaporation on the surface of salt water on the ocean is about 0.8 of that in 
the table. 

The quantity of water evaporated on the surface of all the waters on the earth 
is equal to the quantity of rain-fall. 

Area in Square Miles of the largest Inland Lakes. 


Lakes. 

Eastern Hemisphere 

Aral Sea, Tartary, 

Azov Sea, Russia, . 

Baikal Sea, Siberia, . 
Balkash, Mongolia,. 

Black Sea, Turkey, . 
Caspian Sea, Russia, 
Constance, Switzerland, 
Dead Sea, Palestine, 
Dembia, Abyssinia, 

Enare, Lapland, 

Geneva, Switzerland,. 
Hjelmaren, Sweden, 

Tchad, Africa, 

Ladoga, Russia, 

Loch Lomond, Scotland, 
Lough feaugh, Ireland, 
Onega, Russia, . 
Ouroomia, Persia, . 


Sq. Miles. 

Lakes. 

Sq. Miles 


Tenting, China, 

1200 


Wenern, Sweden, . 

2400 

16650 

Wettern, Sweden, . . 

1045 

8800 

13000 

Zaizan, Mongolia,. 

1600 

5200 

Western Hemisphere. 


. 113000 

Athabasca, N. America, . 

3200 

138000 

Erie Lake, N. America, 

7000 

456 

Great Bear, N. America, . 

4000 

370 

Great Slave, N. America, 

12000 

13000 

Great Salt Lake, 

1880 

870 

Huron, N. America, 

22800 

400 

Maracaibo, S. America, . 

6000 

900 

Michigan, N. America, 

22600 

11600 

Nicaragua, Cent. America, 

3905 

6200 

Ontario, N. America, . 

4950 

27 

Otehenantelcane, N. Amer., 

2500 

80 

Superior, N. America, . 

SOOOO 

3300 

Titicaca, Peru, . 

5400 

1000 

Winnipeg, N. America, 

7200 



















































Difference of Levels. 


361. 


BAROMETRICAL OBSERVATIONS. 

For Determining Difference of Levels. 

Notation of letters for the complete formulse of La Place, in French and English 

measures. 


Lower station 


{ h = height of barometer = A'T 
T = temp, of barometer = T' v 
t = temp, of the air == t'J 


Upper station. 


H — height of barometer at the upper station reduced to the tempei*ature of 
the barometer at the lower station. When the height is read on a brass scale, the 
reduction will be in 

French measures. | English measures. 

H=h f [1+0.0001614 \H=¥ [1+0.00008967 (T-W)]. 

Mean radius of the earth = 6,366,200 metres = 20,886,860 feet. 

Mean height of the atmosphere == 18,336 metres = 60,158.6 feet. 

L — mean latitude between the two stations. 

Z = difference of level between the two stations. 


■ z=l08 ’5 xl8336 x 


Z = log. — x 60158.6 
II 


French measures. 

( 1 + l«j) x ’ ' 

(l + 0.00251 X cos. 2 L) X - 
(l _l 15926 \ 

1 + 6366200 / ' 

English measures. 

(‘ + '- ± W-)x- ■ 

(l + 0.00251 X COS. 2 L) X 
1 Z+_52252\ « 

+ 20886860 / 


( 


The factor (1) gives the difference of level when the observations are made in a 
temperature of 32° Fahr., or 0° Cent., the freezing-point of water, and in latitude 
45°, without the factors of correction (2), (3) and (4). 

The factor (2) is the correction for temperature of the air above or below the 
freezing-point. 

The factor (3) is the correction for latitude above or below 45°. 

The factor (4) is the correction for the decrease of the earth’s attraction. This 
correction is included in the following Table I., to suit any level of the stations. 

There are some other baromeh’ical corrections not included in the above for¬ 
mulae, such as for humidity of the air, capillarity and boiling of the glass tube, 
for the hour of the day and season of the year, all of which are so insignificant, 
uncertain and complicated that I have concluded to omit them. 











Difference op Levels. 




Explanation of tlie Barometrical Tables. 

The tables have been calculated in Peru, under actual practice, by the author. 

Table I. is calculated from the factors (1) and (2), which gives the approximate 
heights above the level of the sea, in English and French measures, for every tenth 
of an hub from 11 to 31 inches. The mean temperature of the air and of the 
barometer is assumed to be 60° Fahrenheit = 15.555 Centigrade, and in latitude 45°. 
The barometer is assumed to be 30 inches = 762 centimetres at the level of the 
sea, but when it is observed to be higher or lower, make the corresponding addi¬ 
tion or subtraction for difference of levels in the table. 

Table II. contains the correction for difference of level in feet or metres at dif¬ 
ferent temperatures of the air above or below 60° Fahr. 

Table III. contains the correction for heights in different latitudes above or 
below 45°. 

Tables IV. and V. are logarithmic corrections for temperature and latitude. 

Table VII, gives the height of a column of air in metres, corresponding to a dif¬ 
ference of one milimetre of mercury at different heights of the barometer. 

Table VIII. gives the height of a column of air in feet, corresponding to a differ¬ 
ence of one-tenth of an inch of mercury at different heights of the barometer. 

Tables X. and XI. contain the correction for the mercurial column at different 
temperatures of the baiometer above or below 60° Fahr. = 15.555 Cent. This 
correction must be made before the barometrical height is applied to Table I. 

Table XII. contains the approximate mean temperature of the air at the level 
of the sea for every month of the year in different latitudes. This table has been 
deduced from observations of Mr. Dove, Humboldt, Raimondi, and other distin 
guished authors. The table agrees very well with the mean temperatures on the 
Atlantic and Pacific coasts, but will not answer for the North Sea and the Baltic, 
where the temperatures are much higher. I have found a great deal of inconve¬ 
nience in the interior of South America for want of a table of this kind. 

When barometrical observations are made far inland, some means must b< 
resorted to for estimating the temperature of the air at the level of the sea in tin 
latitude of observation, in order to make proper corrections for difference of level 
From all the meteorological observations of different authors it appears that the 
mean temperatui’e of 24 successive hours is near 9 o’clock in the morning, and 
that the mean temperature of the day from 9 to 5 p. m. is at noon. 

The variation of temperature throughout the day varies with the latitude, that 
is, the higher the latitude, the greater is the variation. 

Example 1. On the 14th of March, 1869, 2h. 15m. p. m., in Oroya, Peru, lati¬ 
tude 11° 30', the barometer stood 19.46 inches, the temperature of the air 62°, and 
that of the bardfiieter 60°. Required, the height of Oroya above the level of the 
sea in feet ? 


Table I. 


Table XII 


Table II. 


Barometer 19.4 in., 

Correction 0.6 X diff. 142.6 feet, 
Approximate height, 

Temperature at Oroya, 

Latitude 11° 30', 14th of March, 

Mean temp, of the column of air, 

10000 


= 12099.6 feet. 
= 85.5 feet. 

= 12014.1 feet. 
62° Fahr. 

79° 

141 = 70 5°. 


Correct mean temp. 70° 
of the air. 


Table III. Correct for lat. 11° 30', 

Sum of corrections, .... 
Approximate height, . 

Height of Oroya, . 


2000 

10 

'10000 

2000 

10 


feet = 
feet = 
feet = 
feet = 
feet = 
feet = 


208.2 feet. 

41.6 feet. 
0.2 feet. 

23.6 feet. 
4.8 feet. 
0.0 feet. 


278.4 feet. 
12014.1 feet. 

12292.5 feet. 







Aneroid Barometer. 


383 


21.255 inches. 
18.224 inches. 

.006 inches. 
18.218 inches. 


Example. 2. In the city of Paucartarabo, Peru, the barometer was observed to 
stand 21.272 inches, the temperature of the air 70°, and that of the barometer 69°, 
in latitude 13° 18' south. About three miles from.the city, on the mountain 
Huanacaury, the barometer stood 18.224 inches, the temperature of the air 62°, 
and that of the barometer 64°. Required, the height of the mountain above the 
city of Paucartambo? 

Barometer at the lower station, . 21.272 inches. 

Correction for 69°, Table XI., subtract .017 inches. 

Height of barometer at 60°, . 

Barometer at the upper station, 

Correction for 64°, Table XI., subtract 
Height of barometer at 60°, 

Barometer. Heights. 

18.218 13845.0, upper station. 

21.255 9565.3 , lower station. 

Logarithms 3.6314133 = 4279.7 feet, approximate height. 

Table IV. 0.0053929 66° mean temperature. 

Table V. 0.0009888 = 13° 18' latitude. 

3.6377950 = 4343.1 feet, the height required. 

Aneroid Barometer. 

The aneroids made by Negretti & Zambra, London, are compensated, and show 
the height of a column of mercury at the temperature of the freezing-point of 
water, 32° Fahr., or zero Centigrade. The aneroid is not affected by different 
temperatures. When the aneroid is used with the accompanying Table I., a 
correction must be made to convert the column of mercury from 32° to 60° Fahr., 
namely: 


Table I. 


Height of a column of mercury as indicated by the aneroid. 


16 

17 

18 

19 

20 

21 

22 

23 

24 

.25 

26 

27 

28 

29 

30 

.046 

.048 

.050 

.053 

.056 

.059 

.061 

.064 

.067 

.070 

.073 

.075 

.078 

.081 

.084 


Correction in fraction of an inch, always additive. 


Example. Suppose the aneroid to indicate 25.261 inches. 

Correction from the table, . .070 

Height of a column of mercury, 25.331 inches at 60° Fahr. 

Heights of the Principal Mountains and Volcanoes. 


North America. 

Mount St. Elias, 
Mt. Brown, R. M , 
Sierra N evada, Cal., 
Fremont’s Peak, 
Long’s Peak,R. M., 
Cibao Mt., Hayti, 
Cierra del Cobre, 
Black Mt., S. C., 
Mt. Washington. 
Mansfield Mt., Vt., 
Peak of Otter, Vt., 

South America. 

Illimani, Bolivia,* 
Ausangati, Peru,* 
Chimborazo, Eq., 
Sorato, Bolivia, 
l’olima, N. Gran., 
Cerro de Potosi, 
Cerro de Pasco, 
Organ Mt., Brazil, 


Feet. 

17,860 

16.000 

15.500 
13,470 

12.500 
8,600 
7,200 
6,476 
6,234 
4,280 
4,260 


24.100 

22.150 
21,960 
21,500 
18,250 

16.150 
13,780 

7,500 


Europe. 

Elbruz, Caucasus, 
Mont Blanc, Alps, 
Malhaven, Spain, 
Mt. Maladetta, Py., 
Mt. Caballo, Alps, 
Mt. Scardus, Tur., 
Ural Mts., Russia, 
Asia. 

Kunchinginga,Hy. 
Dhawalaghiri, Hy., 
Hindo Koo, Cabul, 
Mt. Ararat, Tur., 
Mt. Lebanon, Syr., 
Africa. 

Abba Yared, A by., 
Piton des Neiges, 
Talba Waha, Aby., 
Oceanica. 

Mt. Ophir, Sum., 
Mt. Semero, Java, 


Feet. 

17,776 

15,668 

11.678 

11,436 

10.154 

10,000 

5,397 

28,176 

28.000 

20,000 

17,210 

12,000 

15,200 

12,500 

12,000 

13,842 

13,000 


Volcanoes, Active. 
Aconcagua, Chili, 
Gualatieri, Peru, 
Cotapaxi, Equador, 
Misti, Peru*,. . 
Popocatapftl, 
Pichincha, Equa., 
Kliutcliewaskaja, 
Vo)can de Fuego, 
Mauna Loa, S. Is!., 
St. Helen’s, Oreg’n, 
Indrapura, Sum., 
Teneriffe, Can. Isl, 
Erebus, Vic. Land, 
Cartago, C. Amer., 
Etna, Sicily, . . 
Hecla, Iceland, . 
Souffriere, Guad., 
Jurollo, Mexico, 
Vesuvius, Italy, 


Feet. 

23,100 

22,000 

19,500 

18,136 

17,735 

16,000 

15,763 

14,000 

13,440 

13.300 

12.300 
12,182 
12.400 
11,480 
10,874 

5,110 

5,108 

4.205 

3,948 


* Measured by the author of this Pocket-book. 





















Barometric and Atmospheric Heights, 


m 


Dif. 


76.59 

75.87 

75.19 

74.56 

73.88 
73.24 
72.63 
71.98 
71.43 
70.77 
70.23 
69 61 
69.07 
68.48 
67.94 
67.42 
66 88 
66.38 
65.84 
65.32 
64.80 
64 34 

63.88 

63.36 

62.91 
62 46 
61.97 
61.50 
61.09 
60.65 
6».23 
59.76 
59.34 

58.92 
58.68 
58.12 
57.79 

57.40 
56.90 

56.57 

56.20 
55 84 
55.47 
55.11 
54.74 

54.41 
54.10 
53.70 

53.37 
53.04 


TABLE I. 


Barometric and Atmospheric Heights. 


French. 


English. 


French. 

s 

Altitude. 

Bar. 

Bar. 

Altitude. 

Dif. 

Dif. 

Altitude. 

1 Bar. 

Bar. 

metres. 

m.m. 

In. 

feet. 

metres. 

m.m. 

in. 

8488 09 

279.4 

ii. 

27848.5 

251.3 

248.9 
246.7 

244.6 

242.4 

240.3 

238.3 

236.2 

234.3 
282.2 

230.4 

228.4 

226.6 

224.7 

222.9 
221.2 

219.4 

217.8 
216.0 

214.3 
212.6 
211.1 

209.6 

207.9 

206.4 

204.9 

203.3 
201.8 

200.4 
199.0 

197.6 

196.1 

194.7 

193.8 

192.5 

190.9 

189.6 

188.3 

187.2 

185.6 

184.4 

183.2 
182.0 

180.3 

179.6 

178.5 

177.5 
176.2 
175.1 
174.0 

52.72 

52.49 

51.94 
51.78 

51.39 

51.14 

50.82 

50.50 
50.20 

49.90 
49.65 

49.34 
49.07 
4S.77 

45.47 

48.18 

47.91 
47.64 

47.40 

47.15 
46 85 

46.60 

46.36 
46.11 
45.88 

45.60 

45.35 
45.10 
44.90 
44.69 
44.44 

44.19 

43.95 

43.74 

43.47 
43.22 
43.00 

42.82 
42.62 
42.42 
42 24 
42.04 

41.82 

41.60 
41.39 

41.15 
40.94 

40.75 
40.54 

40.37 

5318.11 

406.4 

16. 

8411.48 

281.9 

.i 

27597.2 

5265.39 

408.9 

.1 

8335.61 

284.5 

.2 

27348.3 

5212.90 

411.4 

.2 

8260 42 

287 0 

.3 

27101.6 

5160.96 

414.0 

.3 

8185.86 

289.5 

.4 

26857.0 

5109.18 

416.5 

.4 

8111.96 

292.2 

.5 

26614.6 

5057.79 

419.1 

.5 

8038.74 

294.7 

.6 

26374.3 

5006.65 

421.6 

.6 

7966.11 

7894.13 

297.3 

299.8 

.7 

.8 

26136.0 

25899.8 

4955.83 

4905.33 

424.1 

426.7 

.7 

.8 

7822.70 

302.2 

.9 

25665.5 

4855.13 

429.2 

. .9 

7751.93 

304 8 

13. 

25433.3 

4805.23 

431.8 

17. 

7681.70 

307.3 

.1 

25202.9 

4755.58 

434.3 

.1 

7612.09 

309.8 

.2 

24974.5 

4706.24 

4368 

.2 

7543.02 

312.4 

.3 

24747.9 

4657.17 

439.4 

.3 

7474.54 

314.9 

.4 

24523.2 

4608.40 

441.9 

.4 

7406.60 

317.5 

.5 

24300 3 

4559.93 

444.5 

.5 

7339.18 

320.0 

.6 

24079.1 

4511.75 

447.0 

.6 

7272.30 

322.5 

.7 

23859.7 

4463.84 

449.5 

.7 

7205.92 

325.1 

.8 

23641.9 

4416.20 

452.1 

.8 

7140.08 

327.6 

.9 

23425.9 

4368.80 

454.6 

.9 

7074.76 

330.2 

13. 

23211.6 

4321.65 

457.1 

18. 

7009.96 

332.7 

.1 

22999.0 

4274.80 

459.7 

.1 

6945.62 

335.2 

.2 

22787.9 

4228.20 

462.2 

.2 

6S81.74 

337.8 

.3 

22578.3 

4181.84 

464.8 

.3 

6818.38 

340.3 

.4 

22370.4 

4135.73 

467.3 

.4 

6755.47 

342.9 

.5 

22164.0 

4089.85 

469.9 

.5 

6693.01 

345.4 

.6 

21959.1 

4044.25 

472.4 

.6 

6631.04 

347.9 

.7 

21755.8 

3998.90 

474.9 

.7 

6569.54 

350.5 

.8 

21554.0 

3953.80 

477.5 

.8 

6508.45 

353.0 

.9 

21353.6 

3908.90 

480.0 

.9 

6447.80 

355.6 

14. 

21154.6 

3864.21 

482.6 

19. 

6387.57 

358.1 

.1 

20957.0 

3819.77 

485.1 

.1 

6327.81 

6268.47 

360.6 

363.2 

.2 

.3 

20760.9 

20566.2 

3775.58 

3731.63 

487.6 

490.2 

.2 

.3 

6209.55 

365.7 

.4 

20372 9 

3687.89 

492.7 

.4 

6150.87 

368.3 

.5 

20180.4 

3644.42 

495.3 

.5 

6092.75 
6034 96 

370.8 

373.3 

.6 

.7 

19989.7 

19800.1 

3601.20 

3558.20 

497.8 

500.3 

.6 

.7 

5977.56 

3|5.9 

.8 

19611.8 

3515.38 

502.9 

.8 

5920.66 

378 4 

.9 

19425.1 

3472.76 

505.4 

.9 

5864.0) 

381.0 

15 

19239.5 

3430.34 

508.0 

20. 

5807.89 

383.5 

.1 

19055.1 

3388.10 

510.5 

.1 

5752.05 

386.0 

.2 

18871-9 

3346.06 

513.0 

.2 

5696.58 

388.6 

.3 

18689.9 

3304.24 

516.6 

.3 

5641.47 

391.1 

.4 

18509.1 

3262 64 

518.1 

.4 

5586.73 

393.6 

.5 

18329.5 

3221.25 

520.7 

.5 

5532 32 

396.2 

.6 

18151.0 

3180.10 

523.2 

.6 

5478.22 

398.7 

.7 

17973.5 

3139 16 

525.7 

.7 

5424.52 

401.3 

.8 

17797.3 

3098 41 

528.3 

.8 

5371.15 

403.8 

.9 

17622.2 

3057.87 

530.8 

.9 


English. 

Altitude. 

feet. 


Dif. 


17448.2 

17275.2 

17103.3 

16932.6 

16762.7 

16594.1 

16426.3 
16259.6 
16093 9 

15929.2 

15765.5 

15602.6 
1544H.7 

15279.7 

15119.7 

14960.7 

14802.6 

14615.4 

14489.1 

14333.6 

14178.9 

14025.2 

13872.3 

13720.2 

13568.9 

13418.4 

13268.8 
13120.0 
12972.0 

12824.7 

12678.1 

12532.3 

12387.3 

12243.1 

12099.6 
11957.0 

11815.2 

11674.1 

11533.6 

11393.8 

11254.6 
11116.0 

10978.1 

10840.9 

10704.4 

10568.6 

10433.6 

10299.3 

10165.6 

10032.6 


173.0 

171.9 

170.7 
169 9 
168.6 

167.7 

166.7 

165.7 

164.7 

163.7 

162.8 

161.9 
161.0 
160.0 
159.0 

158.1 

167.2 

156.3 

155.5 

154.7 

153.7 

152.9 

152.1 

151.3 

150.5 

149.6 

148.8 
148.0 

147.3 

146.6 

145.8 
145.0 

144.2 

143.5 

142.6 

141.8 

141.2 
1405 

139.8 

139.2 

138.6 

137.9 

137.2 
136.5 
135.8 
135.0 

134.3 

133.7 
133.0 

132.3 


The columns Bar. is the height of the Barometer in inches and milimetres. 

The columns Altitude is the corresponding height of level above the sea in feet 
and metres. 

The altitude in metres can be read from the barometer in inches; or, the altitude 
in feet can be read from the barometer in milimetres. 








































Barometric and Atmospheric Heights, 


36.1 


TABLE I. 

Barometric and Atmospheric Heights. 


Dif. 

40.17 

40.09 

39.81 
39.65 
39.44 
39.23 
39.11 
38.99 

38.74 
38.53 

38.37 
38.-0 
38.00 
37.88 
37.67 
37.52 

37.37 

37.16 
37.06 

30.82 
36.69 

36.59 
36.39 

36.21 
36.09 
35.93 
35.79 

35.60 

35.46 
35 30 

35.21 
35.02 
34.90 

34.75 

34.49 

34.47 
3 4.32 

34.17 
34.05 
33 92 
33.78 

33.61 

33.50 
33.: 7 
83.25 
33.13 
33.01 

32.83 
32.74 

32.61 


French. 


English. 


French. 



English 

Altitude. 

Bar . 

Bar . 

Altitude. 

Dif. 

Dif. 

Altitude. 

Bar . 

Bar . 

Altitude. 

metres. 

m.m. 

in. 

feet. 

metres. 

m.m. 

in. 

feet. 

3017.50 

533.4 

ai. 

9900.1 

131.7 

131.9 

130.6 

130.1 
129.4 

128.7 

128.3 

127.6 

127.1 

126.4 

125.9 

125.3 
124 7 

124.3 

123.6 

123.1 

122.6 
121 9 
121.6 

120.8 

120.4 
129.0 

119.4 
118.8 
1184 

117.9 

117.4 
116.8 
1164 

115.8 

115.5 

114.9 

114.5 
114.0 
113 5 

113.1 

112.6 

112.1 

111.7 

111.3 

110.8 

110.3 

109.9 
109.5 
109.1 

108.7 
108 3 

107.7 

107.4 
107.0 

32.46 

32.34 

32.22 
32.09 
31.98 
31.88 

31.76 
31.64 

31.52 
31.39 
31 27 
31.15 
31.03 
30.94 
30.81 
30.72 
3 i .60 
30.48 
30.41 

30.24 

30.18 
30.05 
29.93 

29.84 
29.75 
29 63 
29 52 
29 45 
29.32 

29.23 
29.11 
29.02 

28.92 

28.84 
28.71 
28.02 

28.53 
28 43 
28 35 

28.25 

28.19 
28.10 
28.01 

27.92 
27.83 

27.77 
27.67 
27.58 
27.52 
27.43 

1210.61 

660.4 

36. 

3971.9 

2977.33 

535.9 

.1 

9768.3 

1178.15 

662.9 

.1 

38 G 5.4 

2137.34 

538.4 

.2 

9637.1 

1145.81 

665.4 

.2 

3750.3 

2897.53 

541.0 

.3 

9506.5 

1113.59 

668.0 

.3 

3653.6 

2 S 57.88 

543.5 

.4 

9376.4 

1081.50 

670 5 

.4 

3548.3 

2318.44 

646.1 

.5 

92 47.0 

1049.52 

673.1 

.5 

3443.4 

2779.21 

548.6 

.6 

9118.3 

1017.64 

675 6 

.6 

3338.8 

2740.10 

551.1 

.7 

89.‘0.0 

985.888 

678.1 

.7 

3234 6 

2701.21 

553.7 

.8 

8862.4 

954.251 

680.7 

.8 

3130.8 

2662.47 

556.2 

.9 

8735.3 

922.734 

6832 

.9 

3027.4 

2623.94 

658.8 

22 . 

8 r. 0 S .9 

891.341 

685 8 

27 . 

2924.4 

2585.57 

561.3 

.1 

8183.0 

860.070 

688.3 

.1 

2821.8 

2547.37 

563.8 

.2 

8357.7 

828 919 

690.8 

.2 

2719.6 

2509.37 

566.4 

.3 

8233.0 

797.891 

693.4 

.3 

26178 

2471.49 

563.9 

.4 

81(18.7 

766 953 

695.9 

.4 

2516 3 

2433 82 

571.5 

.5 

7985.1 

736.140 

698 5 

.5 

2415.2 

2396.30 

574.0 

.6 

78,62.0 

705.416 

701.0 

.6 

2314.4 

2368.93 

576.5 

.7 

7739.4 

674.815 

703.5 

.7 

2214.0 

2321.77 

579.1 

.8 

7617.5 

644 335 

706.1 

.8 

2114.0 

2284.71 

681.6 

.9 

7495.9 

6:3.927 

708.6 

.9 

2014.3 

2247.89 

584.2 

23 . 

7375.1 

583 682 

711.2 

38. 

1915.0 

2211.20 

536.7 

.1 

7251.7 

•'53.503 

713.7 

.1 

1816.0 

2174.61 

589.2 

.2 

7134.7 

523 454 

716.2 

.2 

1717.4 

2138.22 

591.8 

.3 

7015.3 

493 523 

718 8 

.3 

1 C 9.2 

2102.01 

594.3 

.4 

6896.5 

463.683 

721.3 

.4 

1521.3 

2065.92 

696.9 

.5 

6778.1 

433 935 

723.9 

.5 

1423 7 

2029.09 

599.4 

.6 

6660.2 

4< >4.309 

726.4 

.6 

1326.5 

1994.20 

601.9 

.7 

6542.8 

374.785 

728.9 

.7 

1229.6 

1958.60 

604.6 

.8 

6426.0 

345.332 

731.5 

.8 

1133.0 

1923.14 

607.0 

.9 

6309.6 

316.010 

734.0 

.9 

1036.8 

1 S 87.34 

1852.63 

609.6 

612.1 

24. 

.1 

6193.8 

6078.3 

286.788 

257-672 

736.6 

739.1 

39. 

.1 

940.9 

845.4 

1817.61 

614.6 

_2 

5963.4 

228.6)7 

741.6 

.2 

750.2 

1782.71 

617.2 

.3 

5848.9 

199.731 

7442 

.3 

657.3 

1747.96 

619 7 

A 

5734.9 

170.898 

746.7 

.4 

560.7 

1713.37 

622.3 

.5 

5621.4 

142.186 

749.3 

.5 

466.5 

1678.00 

624 .S 

.6 

5508.3 

113.566 

751.8 

.6 

372 6 

1644.58 

627.3 

.7 

5395.7 

85.037 

754.3 

.7 

279 0 

1610.41 

629.9 

.8 

528 7.6 

56.600 

756 9 

.8, 

185.7 

157 6.36 

632.4 

.9 

5171.9 

28.254 

759.4 

.9 

92 7 

1542.44 

635 0 

25 . 

5060.6 

o.oooo 

762.0 

30. 

0.0000 

150 S .66 

637.5 

.1 

4949.8 

28.193 

764.5 

.1 

92.5 

1475.05 

640.0 

.2 

4839.5 

56.295 

767.0 

.2 

184.7 

1441.55 

612.6 

.3 

4729.6 

84.305 

709.6 

.3 

276.6 

1403.18 

645.1 

.4 

4620.1 

112.225 

772.1 

.4 

368.2 

1374.93 

647.7 

.5 

4511.0 

140.053 

7747 

.5 

459.5 

13 11.80 

66,0.3 

.6 

4402.3 

1 G 7.820 

777.2 

.6 

550.6 

1308.79 

652.8 

.7 

4294.0 

195.495 

779.7 

.7 

641.4 

1275 96 

655.3 

.8 

4186.3 

223.079 

782.3 

.8 

731.9 

1243.22 

657.9 

.9 

4078.9 

250.601 

784.8 

.9 

822.2 





278.033 

787 4 

31. 

912.2 


Dif. 

106.5 
106.1 

105.7 
165 3 
104.9 
104 6 

104.2 

103.8 
103.4 
103 0 

102.6 

102.2 

101.8 


100.8 


99.7 

99.3 

99.0 

98.6 

98.2 

97.9 

97.6 

97.2 

96.9 

96.6 

96.2 

95.9 

95.5 

95.2 

94.9 

94.5 

94.2 

93.9 

93.6 

93.3 

93.6 

92.7 

92.5 

92.2 
91.0 

91.6 

91.3 
91.1 

90.8 
90.5 

90.3 
90.0 


The difference in the Bar. m.m. column is 2.5 milimetres ; therefore, multiply the 
difference of altitude in metres by the excoeding milimetres and by 0.4; subtract 
th(5 product from the tabular altitude, and the remainder will be the altitude of 
level in metres, corresponding to the reading of the barometer in milimetres. 


































Correction for Temperature, 


%r, fi 


TABLE II. —Correction for Mean Temperature. 


Ten 

IOihr. 

ip. 

Cent 

1000 

2000 

3000 

Ileig 

4000 

ht in ft 

5000 

set or 

6000 

metre 

7000 

s. 

8000 

9000 

100CO 

Temp. 

Cent, i Fahr. 

61 

J 6.1 

2.08 

4.17 

6.25 

8.34 

10.42 

12.50 

14.59 

16.68 

18.76 

20.85 

15.0 

59 

62 

16.6 

4.17 

8.31 

12.49 

16.61 

20.82 

24.98 

29.15 

33.32 

37.48 

41.65 

14.4 

58 

03 

17.2 

6.25 

12.49 

18.74 

21.98 

31.23 

37.48 

43.72 

49.97 

56.21 

62.46 

13.8'57 

64 

17.7 

8.33 

16.67 

24.99 

33.34 

41.65 

49.98 

58.31 

66.61 

74.97 

83.30 

13.3 

56 

05 

18.3 

10.41 

20.82 

31.24 

41.65 

52.06 

62.48 

72.89 

83.30 

93.72 

104.1 

12.7 

55 

66 

18.8 

12.49 

24.99 

37.48 

49.98 

62.47 

74.96 

87.46 

99.96 

112.4 

124.9 

12.2 

54 

67 

19.4 

14.58 

29.15 

43.43 

58.31 

72.89 

86.S6 

102.1 

116.6 

130.3 

145.8 

11.6 

53 

68 

20.0 

16.66 

33.32 

49.61 

66.64 

83.30 

99.22 

116.6 

133.3 

148.8 

166.6 

11.1 

52 

69 

20.5 

18.74 

37.49 

56.23 

74.98 

93.71 

112.4 

131.2 

149.9 

168.7 

187.4 

10.5 

51 

70 

21.1 

20.82 

41.64 

62.46 

83.2S 

104.1 

121.9 

145.7 

166.5 

187.4 

208.2 

10.0 

50 

. 71 

21.6 

22.91 

45.81 

68.72 

91.63 

114.5 

137.4 

160.3 

183.2 

206.1 

229.1 

9.4 

49 * 

S 72 

22.2 

24.99 

49.98 

74.77 

99.96 

121.9 

149.5 

174.9 

199.9 

224.9 

249.9 

8.8 

48 ~ 

2 73 

22.7 

27.07 

54.14 

SI.21 

108.3 

135.3 

162.4 

189.5 

216.6 

243.6 

270.7, 

8.3 

47 « 

£ 74 

23.3 

29.15 

58.31 

87.46 

116.6 

145.7 

174.9 

201.1 

233.2 

262.4 

291.5 

7.7 

46 <E 

p. 75 23.8 

31.24 

62.47 

93.71 

124.9 

156.2 

187.4 

218.6 

249 9 

281.1 

312.4 

7.2 

-45 e- 

r 76 

24.4 

33.32 

66.64 

99 96 

133.3 

166.6 

199.9 

233.2 

266.5 

299.9 

333.2 

6.6 

44 

+* 77 

25.0 

35.40 

70.80 

106.2 

141.6 

177.0 

212.4 

247.8 

283.2 

318.6 

354 0 

6.1 

43 co 

78 

25.5 

37.48 

74.98 

112.4 

149.9 

187.4 

224.9 

261.4 

299.8 

337.3 

374.8 

5.5 

42 js 

72 79 

26.1 

39.57 

79.12 

118.7 

158.2 

191.8 

237.4 

277.0 

316.5 

356.1 

395.7 

5.0 

41 Z 

3 80 

26.6 

41.65 

83.30 

124.9 

166.G 

208.2 

249.9 

291.5 

333.2 

374.8 

416.5 

4.4 

40 a 

T 81 

27.2 

43.73 

87.46 

131.2 

174.9 

218.6 

262.4 

306.1 

349.8 

393.6 

437.3 

3.8 

39 ? 

2 82 

27.7 

45.81 

91.63 

137.4 

183.2 

229.0 

274.8 

320.7 

366.5 

412.3 

458.1 

3.3 

38 -2 

3 83 

28.3 

47.90 

95.79 

143.3 

191.6 

233.5 

286.6 

335.3 

3S3.2 

431.1 

478.9 

' 2.7 

37 g 

£ 84 

28.8 

49 98 

99.96 

149.9 

199.9 

249.9 

299.8 

349.8 

399.8 

449.8 

499.8 

2.2 

36 £ 

o 85 

29.4 

52.06 

104.1 

156.2 

208.2 

260.3 

312.4 

364.4 416.5 

468.6 

520.6 

1.6 

35 S 

° 86 

30.0 

54.14 

108.2 

162.4 

216.5 

270.7 

324.8 

379.0 433.1 

487.3 

541.5 

1.1 

34 i' 

~ 87 

30.5 

56.23 

112.4 

168.6 

224.9 

281.1 

337.2 

393.6 

449.8 

506.0 

562.2 

0.5 

33 

I 88 

31.1 

58.31 

116.6 

174.9 

233.2 

291.5 

349.8 

408.2 

466.5 

524.8 

583.1 

0.0 

32 5 

~ 89 

31.6 

59.37 

118.7 

178.1 

237.5 

296.8 

356.2 

415.5 474.9 

534.3 

593.7 

—0.5 

31 2 

< 90 

32.2 

61.77 

123.5 

185.3 

247.1 

208.8 

370.6 

432.4 494.1 

555.9 

617.7 

—1.1 

30 3 

91 

32.7 

64.56 

129.1 

193.7 

258.2 

322.8 

387.3 

451.9 516.5 

581.0 

645.6 

— 1.6 

29 m 

92 

33.3 

66.64 

1 00.3 

199.9 

266.5 

333.2 

399.8 

466.5 533.1 

529.7 

666.4 

—2.2 

28 “ 

93 

33.8 

68.72 

137.4 

206.1 

274.9 

343.6 

412.3 

471.0 549.7 

618.5 

687.2 

—2.7 

27 

94 

34.4 

70.80 

141.6 212.4 

283.2 

354.0 

424.8 

495.6 

566.4 637.2 

708.0 

—3.3 

26 

• 95 

35.0 

72.89 

145.8 

218.7 

291.6 

364.4 

437.3 

510.2 583.1 

656.0 

728.9 

—3.8 

25 

96 

35.5 

74.83 

149.6 

224.5 

299.3 

374.1 

449.0 

523.8 598.6 673.5 

748.3 

—4.4 

24 

. 97 

S6.1 

77.05 

154.1 

231.1 

308.2 

385.2 

462.3 

539.3 616.4 1 693.4 

770.5 

—5.0123 

98 

36.6 

79.13 

158.2 

237.4 

316.5 

395.6 

474.8 

553.9 

633.0 

712.2 

791.3 

—5.5 

22 

99 

37.2 

4*1.22 

! 162.4 

-43.6 

324.9 

406.1 

487.3 

560.5 

! 649.7 

731.0 

812.2 

—6.1 

21 

100 

37.7 

83.36 

166.6 

249.9 

333.2 

416.5 

499.8 

583.1 

660.4 749.7 

83-fO 

—6.*> 

20 

Fahr. 

Cent 

1 1000 

; 2000 

3000 

4000 

5000 

6000 

7000 

80001 9000:100' -'0 

Ceut. 

Fahr. 


TABLE III.—Correction for Mean Latitude. 


Mean 

latitude. 


44 
42 
40 
£ 38 
S 36 
H 34 
Z- 30 
S 28 

a) 24 
^ 20 
| 18 
£ 14 

■s io 
* 6 
2 


Heights in feet or metres. 


1000 I 

2000 

3000 

4000 

5000 

6000 

7000 

8000 

9000 

10000 

0.09 

0.18 

0.27 

0.36 

0.45 

0.54 

0.63 

0.76 

0.81 

0.90 

0.27 

0.54 

0.81 

1.08 

1.35 

1.62 

1>9 

2.16 

2.43 

2.7 

0.44 

0.88 

1.32 

1.76 

2.20 

2.64 

3.08 

3.52 

3.96 

4.4 

0.62 

1.24 

1.86 

2.48 

3.10 

3.72 

4.34 

4.96 

5.58 

6.2 

0.79 

1.58 

2.37 

3.16 

3.95 

4.74 

5.53 

6.32 

7.11 

7.9 

0.96 

1.92 

2.88 

3.84 

4.80 

5.76 

6.72 

7.68 

8.64 

9.6 

1.28 

2.56 

3.84 

5.12 

6.40 

7.68 

8.96 

10.2 

11.5 

12.8 

1.43 

2.86 

4.29 

5.72 

7.15 

8.58 

10.0 

11.4 

12.9 

14.3 

1.71 

3.42 

5.13 

6.84 

8.55 

10.26 

12.0 

13.7 

15.4 

17.1 

1.95 

3.90 

5.85 

7.80 

9.75 

11.7 

13.6 

15.6 

17.5 

19.5 

2.06 

4.12 

6.18 

-8.24 

10.3 

12.3 

14.4 

16.5 

18.5 

20.6 

2.25 

4.50 

6.75 

9.00 

11.2 

13.5 

15.7 

18.0 

20.2 

22.5 

2.39 

4.78 

7.17 

9.56 

11.9 

14.3 

16.7 

19.1 

21.5 

23.9 

2.49 

4.98 

7.47 

9.96 

12.4 

14.9 

17.4 

19.8 

22.4 

24.9 

2.54 

5.08 

7.62 

10.2 

12.7 

15.2 

17.8 

20.3 

22.9 

25.4 


Mean 

latitude. 


46 . 
48 ” 
5> a 
52 -2 
54 * 
56 a, 
oO 2 
62 - 
66 £ 
70 

72 « 
76 2 

80 S 

84 « 
88 ^ 
























































































Boiling Water. 


367 


ItABLE IV.—Logarithmic Correction for Temperature of the 

Atmosphere. Always positive. 


Temp. 

Loga- 

Temp. 

Loga- 

Temp. 

Loga- 

Temp. 

Loga- 

Cent. 

Fahr 

ri thins. 

Cent. 

Fahr 

ri thins. 

Cent. 

Fahr 

rithms. 

Cent. 

Fahr 

rithms. 

—2.2 

28 

9.97005 

8.33 

47 

9.98808 

18.8 

66 

0.00539 

29.4 

85 

0.02204 

—1.6 

29 

9.97102 

8.88 

48 

9.98901 

19.4 

67 

0.00628 

30.0 

86 

0.02200 

—1.1 

30 

9.97198 

9.41 

49 

9.98993 

20.0 

68 

0.00717 

30.5 

87 

0.02376 

—0.5 

31 

9.97294 

10.0 

50 

9.99)86 

20.5 

69 

0.00806 

31.1 

88 

0.02461 

0.0 

32 

9.97391 

10.5 

51 

9.99178 

21.1 

70 

0.00895 

31.6 

80 

0.02547 

f 55 

33 

9.97487 

11.1 

52 

9.99270 

21.6 

71 

0.00984 

32.2 

90 

0.02632 

1.11 

34 

9.97682 

11.6 

53 

9.99362 

22.2 

72 

0.01072 

32.7 

91 

0.02717 

1.66 

35 

9.97678 

12.2 

54 

9.99454 

22.7 

73 

0.01 LOO. 

33.3 

92 

0.02802 

2.22 

36 

9.97773 

12.7 

55 

9.99545 

23.3 

74 

0.01248 

338 

93 

0.02886 

2.77 

37 

9.97868 

13.3 

56 

9.99637 

23.8 

75 

0.01336 

34.4 

94 

0.02071 

3 33 

38 

9.97963 

13.8 

57 

9.99728 

24.4 

76 

0.01423 

35.0 

95 

0.03055 

•1.88 

39 

9.98058 

14.4 

58 

9.99819 

25.0 

77 

0.01511 

35.5 

96 

0.03139 

4.44 

40 

9.98152 

15.0 

59 

9.99909 

25.5 

78 

0.01598 

36.1 

97 

0.03224 

51)0 

41 

9.98217 

15.5 

60 

0.00 )00 

26.1 

79 

0.01685 

36.6 

98 

0.03307 

5.55 

42 

9.98341 

16.1 

61 

0.00090 

26.6 

80 

0.01772 

37.2 

99 

0.03391 

6.11 

43 

9.98414 

16.6 

62 

0.00181 

27.2 

81 

0.01859 

37.7 

100 

0.03475 

6 66 

44 

9.98523 

17.2 

63 

0.00270 

27.7 

82 

0.01945 

38.3 

101 

0.03558 

7.22 

45 

9.98622 

17.7 

64 

0.00360 

28.3 

83 

0.02032 

38.8 

102 

0.03641 

7.77 

46 

9.98715 

18,3 

05 

0.00450 

28.8 

84 

0.02118 

39.4 

103 

0.03724 


TABLE V.—Logarithmic Correction for Mean Latitude of 

Observation. Always positive. 


mt. 

Log. 

Lat. 

Log. 

Lat. 

Log. 

Lat. 

Log. 

Lat. 

Log. 

Lat. 

Log. 

0 

0.00111 

15 

0.00096 

30 

0.00055 

45 

0.00000 

60 

9.99944 

75 

9.99904 

1 

0.00110 

16 

0.00094 

31 

0.00052 

46 

9.99996 

61 

9.99941 

76 

9.99902 

2 

0.00110 

17 

0.00992 

32 

0.00048 

47 

9.99992 

62 

9.99938 

77 

9.99900 

• > 

O 

0.00110 

18 

0.00089 

33 

0.00045 

48 

9.99988 

63 

9.69935 

78 

9.998 >8 

4 

0.00109 

19 

0.00087 

34 

0.00041 

49 

9.99984 

64 

9.99932 

79 

9.99897 

5 

0.00109 

20 

0.00085 

35 

0.00038 

50 

9.99981 

65 

9.99929 

80 

9.99896 

6 

0.00108 

21 

0.00082 

36 

0.00034 

51 

9.99277 

66 

9.99923 

81 

9.99894 

7 

0.00107 

22 

0 00079 

37 

0.00030 

52 

9.99973 

67 

9.99923 

82 

9.29 93 

8 

0.00106 

23 

0.00077 

38 

0.00027 

53 

9.99969 

68 

9.99920 

S3 

9.99892 

9 

0.00105 

24 

0.00074 

39 

000023 

54 

9.99963 

69 

9.99917 

84 

9.998J1 

10 

0.00104 

25 

0.00071 

40 

0.00019 

55 

9 99962 

70 

9.99915 

85 

9.99891 

11 

0 00103 

26 

0.00068 

4L 

0.09015 

56 

9.99958 

71 

9.99913 

86 

9.99890 

12 

0.00101 

27 

0.00965 

42 

0.00011 

57 

9.99955 

72 

9.99910 

87 

9.99889 

13 

0.00099 

2S 

0.00062 

43 

0.00008 

58 

9.99951 

73 

9.99908 1 

88 

9.99889 

14 

0 0009S 

29 

0.00059 

44 

0.00004 

59 

9.99948 

74 

9.99306 1 

89 

9.99889 


TABLE VI.—Temperature of Boiling “Water, 

Corresponding to the Height of the Barometer at 60° Fahrenheit. 


French Measures. 

Height Temp. 
Diff. Barom. Water 
M. M. Cent. 

434.07 85.00 

441.68 85.55 

453.45 86.11 

463.41 86.66 

473.55 87.22 

483.86 87.77 

4 14.33 88.33 

504 99 8 699 

51584 89.44 

526.92 90.00 

53616 90.55 

549 62 91.11 

561.26 91.66 

573.11 92.22 

585.18 92.77 


9.61 
9 77 
9 96 
0.14 
10.31 
lit 47 
10 66 
10 85 
LI.08 
11.24 
11.46 

11.84 

11.85 
12 07 
12 27 


English Measures. 

Temp. Height 
Water Barom. Diff. 
Fahr. Inches. 

185 17.090 

186 17.468 

187 17.853 

188 18 245 

189 18.644 

190 19.050 

191 19.462 

192 19.682 

193 20.309 

194 20.745 

195 21.188 

196 21.639 

197 22.097 

198 22.564 

199 23.039 


.378 
.385 
.392 
.399 
.406 
.412 
.429 
. 27 
.436 
.443 
.451 
.458 
.467 
.475 
.483 


French Measures. 

Height Temp. 
Barom. Water. 
M. M. Cent. 

597.45 93.33 

609.94 93.88 

622.64 94.44 

635.55 95.60 

648.73 95.55 

662.09 96.11 

675.66 96.66 

689.49 9r.22 
703.54 97.77 

717 82 98.33 

732.35 98 88 

747 13 9 1.44 

762.17 100.0 
777.43 If(0.5 

792.95 101.6 


Diff. 

12.49 

12.70 

12.91 

13.18 

13.36 

13.57 

13.83 

14.05 

14.28 

14.53 

1478 

15.04 

15.26 

15.52 


English Measures. 

Temp. Height 
Water Barom. 

Fahr. Inches. 

200 23.522 

201 24.014 

202 24.514 

203 25.022 

204 “25,551 

205 28.067 

206 26.6' i2 

207 27.146 

2 8 27.639 

209 28 261 

210 28.833 

211 29.415 

212 30.007 

213 30.60S 

214 31.219 


Diff. 

.492 

.500 

.508 

.519 

.526 

.535 

.544 

.553 

.562 

.572 

.582 

.592 

.691 

.611 
















































































£8S Columns of Air and Mercury. 



TABLE VII 

Corresponding to 

.—Height of a Column of Air in Metres, 

one milimetre in the barometer , at different temperatures. 


Bar. 



Centigrade temperature of the air and the barometer. 




M.M. 

—6 

—3 

0 


+3 

6 

9 

12 

l ^ 

18 

21 

! 24 

27 

SO 


33 

36 

400 

20.7 

20.8 

20.9 

21.0 

21.1 

21.2 

21.3 

21.4 

21.5 

21.6 

21.7 

21.8 

21.9 

22.0 

22.1 

420 

19.7 

19.8 

19.9 

20.0 

20.1 

20.2 

20.3 

20.4 

120.5 

20.6 

20.7 

20.8 

20.9 

21.0 

21.1 

440 

18.S 

18.9 

19.0 

19.1 

19.2 

19.3 

19.4 

19.5 

19.6 

19.7 

19.8 

19.9 

20.0 

20.1 

20.2 

460 

18.0 

18.1 

18.2 

18.3 

18.4 

18.4 

18.5 

18.6 

18.7 

18.8 

18.8 

18.9 

19.0 

19.1 

19.2 

480 

17.2 

17.3 

17.4 

17.5 

17.6 

17.6 

17.7 

17.8 

17.9 

18.0 

18.1 

18.1 

18.2 

18.3 

18.4 

500 

16.5 

16.6 

16.7 

16.8 

16.9 

16.9 

17.0 

17.1 

17.2 

17.3 

17.3 

17.4 

17.5 

17.6 

17.7 

520 

15.8 

15.9 

16.0 

16.1 

16.2 

16.2 

16.3 

16.4 

16.5 

16.6 

16.6 

16.7 

16.8 

16.9 

17.0 

540 

15.3 

15.3 

15.4 

15.5 

15.6 

15.6 

15.7 

15.8 

15.9 

16.0 

16.0 

16.1 

16.2 

16.3 

16.3 

560 

14.8 

14.8 

14.9 

15.0 

15.1 

15.1 

15.2 

15.3 

15.4 

15.5 

15.5 

15.6 

15.7 

15.8 

15.8 

580 

14,3 

14.3 

14.4 

14.5 

14.G 

14.6 

14.7 

14.8 

14.9 

15.0 

15.0 

15.1 

15.2 

15.3 

15.3 

GOO 

13.8 

13.8 

13.9 

14.0 

14.1 

14.1 

14.2 

14.3 

14.4 

14.5 

14.5 

14.6 

14.7 

14.8 

14.8 

G20 

13.3 

13.3 

13.4 

13.5 

13.6 

13.6 

13.7 

13.8 

13.9 

14.0 

14.0 

14.1 

14.2 

14.3 

14.3 

040 

12.9 

12.9 

13.0 

13.1 

13.2 

13.2 

13.3 

13.4 

13.5 

13.6 

13.6 

13.7 

13.8 

13.9 

13.9 

660 

12.5 

12.6 

12.6 

12.7 

12.8 

12.8 

12.9 

13.0 

13.1 

13.2 

13.2 

13.3 

13.3 

13.4 

13.4 

680 

12.2 

12.2 

12.3 

12 3 

12.4 

12.5 

12.5 

12.6 

12.7 

12.7 

12.8 

12.9 

12.9 

13.0 

13.1 

700 

11.9 

11.9 

12.0 

12.0 

12.1 

12.2 

12.2 

12.3 

12.4 

12.4 

12.5 

12.6 

12.6 

12.7 

12.7 

720 

11.5 

11.5 

11.6 

11.6 

11.7 

11.8 

11.8 

11.9 

11.9 

12.0 

12.1 

12.1 

12.2 

12.3 

12.3 

740 

11.2 

11 2 

11.3 

11.3 

11.4 

11.5 

11.5 

11.6 

11.7 

11.7 

11.8 

11.9 

11.9 

12.0 

12.0 

760 

10.9 

10.9 

11.0 

11.1 

11.1 

111.2 

11.2 

11.3 

11.4 

11.4 

11.5 

11.6 

11.6 

11.7 

11.7 

780 

10.6 

10.6 

10.7 

10.8 

10.8 

110.9 

10.9 

11.0 

11.1 

11.1 

11.2 

11.3 

11.3 

11.4 

11.4 


TABLE VIII.- 

-Height of 

a Column of Air in 

Feet 

9 


Corresponding to one-tenth of an inch in the barometer at different temperatures. 

Bar. 



Fahrenheit temperature of the air and the barometer. 




In. 

30° 

35° 

40° 


45° 

50° 

55° 

60° 

65° 

70° 

75° 

80° 

85° 

90° 

95° 

100° 

16 

163 

165 

167 


168 

170 

172 

174 

176 

173 

179 

181 

183 

185 

187 

188 

17 

153 

155 

156 


158 

159 

161 

163 

165 

166 

168 

170 

171 

173 

175 

177 

18 

145 

157 

158 


160 

161 

162 

154 

156 

157 

159 

160 

162 

163 

165 

166 

19 

135 

137 

138 


140 

142 

144 

146 

148 

149 

151 

152 

153 

155 

157 

158 

20 

130 

132 

133 


135 

136 

137 

139 

140 

142 

143 

145 

146 

118 

149 

151 

21 

124 

126 

127 


128 

130 

131 

132 

133 

135 

136 

137 

139 

140 

142 

143 

*22 

118 

120 

121 


123 

124 

125 

126 

127 

129 

130 

131 

132 

134 

135 

136 

23 

112 

114 

115 


116 

117 

118 

120 

121 

122 

124 

125 

126 

127 

129 

130 

24 

H)8 

110 

111 


112 

113 

114 

115 

116 

117 

119 

120 

121 

122 

123 

125 

25 

104 

106 

107 


108 

109 

110 

111 

112 

113 

115 

116 

117 

118 

119 

120 

25.5 

102 

104 

105 


106 

107 

108 

109 

110 

112 

113 

114 

115 

116 

117 

118 

26 

100 

102 

103 


104 

105 

106 

107 

10 3 

109 

111 

112 

113 

114 

115 

116 

26.5 

98 

100 

101 


102 

103 

101 

105 

106 

108 

109 

110 

111 

112 

113 

114 

27 

96 

97 

98 


100 

101 

102 

103 

104 

105 

107 

108 

109 

110 

111 

112 

27.5 

95 

96 

97 


98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

28 

93 

94 

95 


96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

28.5 

91 

92 

93 


94 

95 

\ 96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

29 

90 

91 

92 


93 

94 

95 

96 

9, 


98 

99 

100 

101 

102 

103 

104 

29.5 

88 

89 

90 


91 

92 

1 93 

94 

9' 


96 

97 

98 

99 

100 

101 

102 

30 

87 

88 

89 


90 

91 

* 92 

93 

94 

9) 

96 

97 

98 

99 

100 

101 



TABLE IX 

—Mean Height of the Barometer 




in different countries , reduced to the level of the 

sea, and to 66° Fahr. 

temperature. 






Inches. 




luches. 






Inches. 

Africa, Northern, . 


30.26 

China, 

. 


30.11 

Peru, 




30.09 

Atlantic coast , K. A., 




Denmark, 



29.99 

Prussia 





30.00 

Northern States, 


30.10 

England, 

. 


30.03 

Scotland, . 



29.93 

Southern States, 


30.17 

France, . 

, 


30.00 

Sicily, 




30.11 

Australia, 

. 

• 


30.00 

Greenland, 


29.75 

Spitzbergeu, 



29.87 

Brazil. . 




30.15 

Italy 


. 


30.00 

Sweden 




29.90 

Canary Islands, . . 


30.16 

Iceland, 



29.70 

Venezuela, . 



30.00 

Cape Good Hope, . 


30.11 

Norway, . 



29.89 

West In. lslauds, 


30.02 
















































































Correction for the Barometer. 


369 


TABLE X.—Correction for tlie Mercurial Column 


in Millimetres at Different Temperatures of Barometer. 


Tem. 




Height of barometer in millimetres. 



| 

Temp, 

Cea. 

415 

440 

465! 490 

515 

540 

565 

590 

615 

640 

665 

690 

715 

740 

765 

Cent. 

• 16 

0.03,0.05 

0.04 0.04 

0.04 

0.04 

0.04 

0.05 

0.05 

0.05 

0.05 

0.05 

0.06 

0.06 0.06 

15 

& 1 i 

,0.10,0.11 

0.11 0.12 

0.12 

0.13 

0.14 

0.14 0.15 

0.15 

0.16 

0.17 

0.17 

0.18 0.18 

14 „■ 

5 18 

0.16 

0.1S 

0.19,0.20 0.21 

0.22 

0.23 0.24 

0.25 

0.26 

0.27 

0.28 

0.29 

0 80 

0.30 

i3 e 

§19 

0.24 

0.25 

0.26 0.28 

0 29 

0.30- 

0.32 

0.33 

0.35 

0.36 

0.38 

0.39 

0.40 

0.42 

0.13 

12 £ 

£20 

0.35 

0.37 

0.39 0.42 

0.44 

0.46 

0.48,0.50 

0 52 

054 

0 56 

0.59 

0.61 

0.63 

0.65 

n g 

§21, 

10.42 

0.45 

0.42 0.51 

0.53 

0.56 

0.59 

0.61 

064 

0.66 

0.69 

0.71 

0.74 

0.76 

0.78 

10 £ 

» 22 

0.49 

0.52 

0.55,0.58 

0 60 

0.63 

0.66 

0.69 

0.72 

0.75 

0 78 

0.81 

0.84 

0.87 

0 90 

9 I 

| 23 

0.55 

0.5S 

0.62 0.65 

0.69 

0.72 

0.75 

0.79 

0.82 

0.85 

0.89 

0.92 

0.95 

0.99 

1.02 

8 r 

r 24 

0.62 

0.66 

0.70 

0.73 

0.77 

0.81 

0.85 

0.88 

0.92 

0.96 

1.00 

1.03 

1.07 

1.11 

1.15 

7 5 

£ 25 

0.09 

0.73 

0.77 

0.81 

0 85 

0.89 

0.94 

0.98 

1.02 

1.06 

1.10' 

1.14 

1.19 

1.23 

1.27 

6: 

g 26 

0.75 

0.80 

0.84 

0.89 

0.93 

0.98 

1.02 

1.07 

1.11 

1.16 

1.20 

1.25 

1.30 

1.34 

1.39 

5 <2 

'§ 27 

0.82 

0>,7 

0.92 

0.97 

1.02 

1.08 

1.12 

1.17 

1.22 

1.27 

1.32 

1.37 

1.42 

1.47 

1.52 

4 g 

a 28 

0.89 

0.94 

1.00 

1.05 

1.10 

1.16 

1.21 

1.26 

1.32 

1.37 

1.42 

1.48 

1.53 

1.59 

1.64 

3 1 

§20 

0.98 

1.0! 

1.07 

1.13 

1.19 

1.24 

1.30 

1.36 

1.41 

1.47 

1.53 

1.59 

1.64 

1.70 

1.76 

2 h 

-30 

1.02 

1.09 

1.15 

1.21 

1.27 

1.33 

1.40 

1.46 

153 

1.58 

1.64 

1.70 

1.76 

1.83 

1.89 

1 g 

- 31 

1.09 

1.16 

1.22 

1.29 

1.85 

1.42 

1.48 

1.55 

1.61 

1.68 

1.74 

1.81 

1.88 

1.94 2.01 

0 - 

§32 

1.16 

1.23 

1.30 

1.36 

1.43 

1.50 

1.56 

1.64 

1.71 

1.78 

1.85 

1.92 

1.99 

2.06 2.13 

—1 ^ 

-fe 33 

1.23 

1.30 

1.38 

1.45 

1.52 

1.60 

1.67 

1.74 

1.82 

1.89 

1.97 

2.04 

2.11 

2.19 

2.26 

—2 £ 

1 34 
35 

1.29 

1.37 

1.44 

1.52 

1.60 

1.68 

1.76 

1.83 

1.91 

1.99 

2.07 

2.11 

2 22 

2.30 2.38 

—3 < 

1.36 

1.44 

1.52 

1.60 

1.69 

1.7611.85 

1.93 

2.01 

2.10 

2.17 

2.25 

2.33 

2.42 2.50 

—4 


TABLE XI.—Correction for tlie Mercurial Column in Thou- 

sands of an Inch , at Different Temperatures of the Barometer above or belozv 60°. 


Temp. 

Falir. 

16 

17 

11 

18 

eigh 

19 

t, of 
20 

the l 
21 

>aror 

22 

ne tel 
23 

in i 
24 

nche 

25 

S. 

26 

27 

28 

29 

30 

Tern] 

Falir. 

d 

62 

003 

003 

003 

003 

003 

003 

004 

004 

004 

004 

005 

005 

005 

005 

005 

58 

a 

64 

006 

006 

006 

007 

007 

007 

008 

008 

008 

000 

0< 9 

009 

010 

010 

010 

56 

o 

66 

009 

009 

009 

010 

011 

011 

012 

012 

012 

013 

014 

014 

015 

015 

016 

54 a 


68 

011 

012 

013 

013 

014 

015 

016 

016 

017 

018 

018 

019 

020 

020 

021 

52 | 

C 

70 

014 

015 

016 

017 

018 

019 

020 

020 

021 

022 

023 

024 

025 

026 

026 

50 ° 

p 

72 

017 

018 

019 

020 

021 

022 

023 

024 

024 

027 

028 

029 

030 

031 

032 

48 .5 

o> 

74 

020 

"21 

022 

024 

025 

026 

027 

028 

029 

031 

032 

034 

035 

036 

038 

46 g 

o 

76 

023 

021 

026 

027 

028 

030 

031 

032 

034 

036 

037 

039 

040 

041 

043 

44 g 

2d 

*-> 

78 

026 

027 

029 

030 

032 

033 

035 

036 

038 

040 

042 

044 

045 

046 

048 

42 ~ 

a 

80 

029 

031 

032 

034 

036 

037 

039 

041 

043 

045 

046 

048 

050 

052 

054 

40 £ 

C*-l 

82 

031 

033 

035 

037 

039 

041 

043 

045 

048 

049 

051 

053 

055 

057 

059 

38 c 

d 

o 

84 

034 

036 

039 

041 

043 

045 

047 

049 

052 

054 

056 

058 

060 

062 

064 

36 c 

D 

86 

037 

010 

042 

044 

046 

049 

051 

053 

056 

058 

060 

063 

065 

067 

070 

34- 

p 

88 

049 

043 

045 

047 

050 

052 

055 

057 

061 

063 

065 

068 

070 

072 

075 

32 §5 

O 

o 

90 

043 

046 

048 

051 

054 

056 

059 

061 

065 

067 

070 

072 

075 

077 

080 

30 § 

o 

92 

046 

049 

051 

054 

057 

060 

063 

065 

069 

071 

074 

077 

080 

083 

086 

28 l 


94 

049 

052 

054 

057 

060 

064 

067 

069 

074 

076 

079 

082 

085 

088 

091 

26 = 

o 

ctj 

96 

051 

055 

058 

060 

063 

067 

071 

073 

078 

080 

084 

087 

090 

093 

097 

24 S 

-t-3 

98 

054 

058 

061 

064 

067 

071 

075 

077 

082 

085 

088 

092 

095 

098 

102 

22 < 

d 

100 

057 

061 

064 

068|071 

075 

079 

082 

086 

089 

093 

097 

100 

104 

107 

20 


Heights in Feet of tlie Principal Waterfalls. 


Gavarny, Pyrenees, 

1260 

Gray Mare’s Tail, 

350 

Rupin, Himalayas, 

120 

Lauterbrun, Switz.. 

912 

Hepste, 

300 

Kakabika, S. Am., 

115 

Staubbach, Switz., 

900 

Nakchikin, Kamcli. 

300 

Lidford, England, 

100 

Ruiean, Norway, 

800 

Terni, Italy, 

270 

Genesee, N. York, 

100 

Seculego, Pyrenees, 

795 

Montmorency, Can., 

242 

Ovapock, S. Amer., 

80 

Lulea, Sweden, 

600 

Foyers, Scotland, 

207 

Rhine Lauffen, Swl. 

65 

Tequendama,Colum. 

540 

Wilberforce, N. A., 

160 

Trollhetta, Sweden, 

60 

Tosa. Piedmont, , 

470 

Cetina, Dalmatia, 

150 

Parana, Paraguay,. 

52 

Missouri, N. Amer., 

400 

Niagara Falls, 

145 

Tivoli, Italy, 

50 

Powerscaurt, lrel., 

380 

Tendon, France, 

125 

Cataracts of Nile, 

40 


24 






























































































Temperatures. 


570 


' 

TABLE XII.—Mean Temperature ol* the Air 

at the Level of the Sea. 


Months in the 


North Latitude. 



South Latitude. 



year. 

60 

50 

40 

30 

20 

10 

0 

10 

20 

30 

40 


January, 

25 

46 

62 

72 

78 

80 

80 

80 

77 

75 

72 


February, 

28 

48 

63 

73 

78 

81 

80 

80 

77 

74 

71 


March, 

32 

50 

64 

74 

79 

82 

81 

79 

76 

72 

69 


April, 

38 

55 

67 

76 

81 

S3 

82 

79 

75 

70 

64 


May, . 

48 

61 

72 

78 

83 

84 

83 

78 

74 

68 

61 


June, 

58 

67 

75 

80 

84 

85 

84 

78 

72 

66 

65 


July, . 

61 

69 

76 

81 

85 

86 

84 

77 

73 

61 

52 

<D 

August, . 

59 

68 

75 

80 

84 

85 

83 

78 

73 

64 

51 

A 

September, . 

52 

64 

72 

78 

83 

84 

82 

78 

72 

62 

£4 

& ! 

October, . 

44 

57 

68 

76 

81 

S3 

81 

79 

71 

63 

59 


November, . 

35 

52 

65 

74 

80 

82 

80 

79 

73 

66 

66 


December, 

28 

48 

63 

73 

79 

81 

80 

79 

75 

71 

71 



January, 

—3.8 

7.7 

16.6 

22.2 

25.5 

26.6 

26.6 

26.4 

25.3 

23.8 

22.2 

February, 

—2.2 

8.8 

17.2 

22.7 

25.8 

27.2 

26.8 

26.6 

25. 

23.3 

21.6 

March, 

0.0 

10 . 

17.7 

23.3 

26.1 

27.7 

27.2 

26.1 

24.4 

22.2 

20.5 

April, 

+3.3 

12.7 

19.4 

24.4 

27.2 

28 3 

27.7 

26.1 

23.8 

21.1 

17.7 

May, . 

8.8 

16.1 

22.2 

25.5 

28.3 

28.8 

28.3 

25.5 

23.3 

20 . 

16.1 

June, 

14.4 

19.4 

23.8 

26.6 

28.8 

294 

28.8 

25.5 

22.2 

18.8 

18.3 

July, . 

161 

20.5 

24.4 

27.2 

29.4 

30. 

28.8 

25. 

22.7 

17.7 

11.1 

August, . 

15. 

20 . 

23.8 

26.6 

28.8 

29.4 

28.3 

25.5 

22.7 

17.7 

10.5 

September, . 

11.1 

17.7 

22.2 

25.5 

28.3 

28.8 

27.7 

25.5 

22.2 

16.6 

12.2 

October, . 

6.6 

13.8 

20 . 

24.4 

27.2 

28.3 

27.2 

25.8 

21.6 

17.2 

15. 

November, . 

1.6 

ll.l 

18.3 

23 3 

26.6 

27.7 

26.8 

26. 

22.7 

18.8 

18.8 

December, 

—2.2 

8.8 

17.2 

22.7 

26.1 

27.2 

26.6 

26.1 

23.8 

21.6 

21 + 


Heights of Natural and Artificial Works. 


Heights Above Level of the Sea. 

Feet. 

Heights Above the Ground. 

Feet. 

Green in a balloon, 1837, . 

27.000 

Tower of Babel, said to have been 

680 

Gav-Lussac, Paris, 1804, 

22,900 

Pyramid Cheops, Egypt, . 

520 

Highest flight of condor, 

21,000 

Tower of Baalbee, Syria, 

500 

Humboldt in the Andes, 

19,500 

St. Peter’s Cathedral, Borne, . 

500 

Growth of vegetation, . 

17,000 

Spire of Strasbourg, 

486 

The author in the Andes,* 

15.120 

Cathedral, Antwerp, 

47 C 

Lake Manasarooa, Thibet, 

14.500 

St. Stephen’s spire, Vienna, 

465 

Pine ami birch grow. 

14.000 

Highest chimney, Glasgow, 

455 

Highest habitation of people,* 

14000 

Spire of Salisbury, 

450 

Potosi silver mine, Bolivia, 

13350 

Cathedral, Milan, 

438 

Lake Titicaca, Peru,* . 

13.000 

St. Mary, Liibeck, . 

404 

La Paz, Bolivia,* 

12-400 

Cathedral, Florence,. 

384 

Poplar grows at 

12000 

St. Paul, London, . 

366 

City of Cuzco, Peru,* 

11,500 

Hotel des Invalides, Paris, 

344 

Oak grows at . 

11,000 

Cathedra], New York, . 

325 

City Riobamba, Andes, 

10-800 

Dome of Capitol, Washington, 

287 

Quito, Equador, 

9.560 

Trinity Church, New York, 

286 

City St. Bernard, Switzerland, 

8,600 

Notre Dame, Paris. . 

220 

City Santa Fe de Bogota, 

8.350 

Column City of London, 

202 

Wild monkeys found at* . 

8.000 

Porcelain. China, 

200 

City of Mexico, 

6,990 

Leaning Tower of Pisa, 

188 

St. Gothanl. Alps, 

6.900 

Alexander Column, St. Petersb’g, 

175 

Lake Lucon, France, 

6.220 

July Column, Paris, 

157 

Palm and bananas grow at 

2.500 

Column Napoleon, Paris,. 

13S 


* Measured by the author of this Pocket-book. 































































L. Johnson & Co.’s Proportions of Tyte. 


371 


fiiam. Pearl. Agate. Nonp. 


1 

2 
S 

4 

5 

6 

7 

8 
9 

10 

1 

2 
3 

4_ 

5 “ 

6 

8 

8 

9 

20 

1 

2 

3 

4 

5 

6 

7 

8 
9 

- 30 - 

1 

2 

3 

4 

5 

6 

7 

8 
9 

40 

1 

2 

3 

4 

6 

7 

8 
9 

50 

1 

2 

3 

4 

5 

6 

7 

8 
9 

- 60 - 

1 

2 

3 

4 

5 

6 

7 

8 
9 

70 

1 

2 

3 

4 
5 _ 
6 

7 

8 
9 

80 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

1 

2 

3 

“ 4' 

5 

6 

7 

8 
9 

20 

1 

2 

3 

4 

5 

6 

- 7. 

8 

9 

30 

1 

2 

3 

4 

5 

6 

7 

8 
9 

40 

- 1. 

2 

3 

4 

5 

6 

7 

8 
9 

50 

1 

2 

4 

4 

- 5. 
6 

7 

8 
9 

60 

1 

2 

3 

4 

5 

6 

7 

8 


1 
2 

3 

4 

5 

6 

7 

8 
9 

10 
11 
- 12 - 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 
23 

.24- 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
36_ 

"37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 
-49“ 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 
-61- 

70 62 

1 63 

\ 64 

O 


Min. Brevier Bourg. 
1 1 1 
2 

3 

4 

5 

6 

7 

8 

- 9- 
10 
11 


2 

3 

4 

5 

6 

7 

8 

9 

-lO- 

11 

12 


13 12 

14 13 


15 

16 

17 

18 
19 
- 20 - 
21 
22 
23 


24 22 

25 23 


26 
27 25 


2 

3 

4 

5 

6 

7 

8 

‘ 9- 
10 
11 
12 

13 

14 

15 

16 
17 
18" 

19 

20 
21 
22 

24 23 


14 

15 

16 
17 
.18. 

19 

20 
21 


28 

29 

-30- 

31 

32 

33 


26 

27 

'28' 

29 

30 


34 31 

35 32 


33 

34 

35 

36 

"37“ 

38 

39 

40 38 

41 39 


24 

25 
-26- 

27 

28 

29 

30 

31 

32 

33 

34 
-35- 

36 

37 


36 

37 

38 

39 
- 40 - 

41 

42 

43 

44 

45 

46 

47 

48 

49 

-50 4 6 u- 

51 47 44 

52 48 


42 

43 


40 

41 


44 42 

45 43 


53 


45 

46 


L. Prim. 
1 
2 

3 

4 

5 

6 

7 

" 8“ 

9 

10 

11 

12 

13 

14 

_ 15 _ 

16 

17 

18 

19 

20 
21 
22 

- 23 - 

24 

25 

26 

27 

28 
29 
30 _ 

"31 

32 

33 

34 

35 

36 

37 

- 38 - 

39 

40 


8. Pica. Pica. 

1 1 


English Gr.Prim. 


2 

3 

4 

5 

6 


2 

3 

4 

5 


7 — 6 . 
8 7 


9 

10 


8 

9 


11 10 

12 


13 

- 14 - 

15 


11 

12 

13 


16 14 

17 15 

18 16 

19 17 


18 

19' 


20 

- 21 —: 

22 

23 20 

24 21 

25 22 

26 23 
27 24 

-28—97 

29 ^ 

30 26 

31 27 

32 28 

33 29 

34 80 
•35—31- 
36 


1 

2 

3 

4 

5 
6 " 
7 


1 

2 

3 

4 

5 


6 

7 

8 
9 


37 


32 


8 

9 
10 
- 11 - 

12 10 

d n 
15 12 
1613 

17 14 

18 7 * 

19 15 

20 16 
21 17 

fr 18 

24 19 

25 20 

20 21 
27 99 

~28~ff 











372 


Belting and Pulleys. 


BELTING. 

Fig. 1, Plate III., represents a pulley hung from the points a and b by the 
belt T, t, forming an angle of contact 2 z on the pulley. The weight IF is hung 
on the pulley for stretching the belt from a and b, in which case the tensions 
Tand< will be alike. The letters on the illustrations correspond with the 
letters in the formulas, and the number of each example corresponds with 
the number of the formula used. 

W= weight in pounds hung on the belt. 

Tand t = respective tensions of the belt in pounds. 

if=half the angle of contact of the belt on the pulley. 

C = force of contact of the belt on the pulley in pounds. 

F= motive force in pounds transmitted by the belt. 

/= friction coefficient of the surfaces in contact. 

F— ( T — t). T={F+t). t = (T — F). T+t=F+2t = 2T-F. 

Fig. 2, Plate III., represents two pulleys of different sizes and connected 
by a belt T, l, in which case the smallest pulley will be in the same condition 
as that in Fig. 1, but the weight IF is the pressure in the journal boxes, and 
the system is arranged for transmitting power. The greatest motive force 
Fthat can be transmitted by the belt cannot exceed the product of the force 
of contact C and the friction / without slipping of the belt. 

When jF>/C, the belt will slide. 

When F~=.fC\ there is no sliding. 

In good practice the motive force Fshould not exceed 75 per cent, of fC. 

Example 1. Fig. 1. The weight IF= 84 pounds, and half the angle of con¬ 
tact Z= 45°. Required the sum of tension. 

Tension T-\- t = ^ = 118.7928 pounds, 

sin. 45° 0.70711 


That is, 118.7928 :2 — 59.3964 pounds, the tension at each point of suspension 
a and b. 

Example 2. It is found by experiment that the tension at a is 41.36 pounds, 
that is Tt — 41.36 X 2 =*82.72 pounds, half the angle of contact being Z= 
48° 36'. Required the weight IF. 

Weight IF— 82.72 X sin. 48° 36' = 82.72 X 0.75011 = 61.05 pounds. 
Example 4. The suspended weight 1F= 86 pounds and the angle of contact. 
120°, making Z— 60°. Required the force of contact of the belt on the pulley. 


86 X 3.1416 X 60 
180 X sin.60° 


104 pounds. 


Example 9. Fig. 2. What motive force F can be transmitted by a leather 
belt of tension T + t = 45 pounds when at rest, half the angle of contact Z= 
78° on a smooth cast-iron pulley of friction/= 0.35 ? 


Motive force F— 


0.35 X 3.1416 X 78 X 450 
180 


214.4 pounds. 


This is the maximum motive force that can be applied under the given con¬ 
ditions, but only 75 per cent, of it should be applied in practice, or 214.4 X 
0.75 = 160.8 pounds. 

When at rest, (T+ t) — 450 pounds, or T— t = 225 pounds, but when in mo¬ 
tion half the motive force is added to T— 250 + 107.2 = 357.2 pounds, the 
pulling tension, and the other half of the motive force is subtracted from t = 
225 —107.2 = 117.8 pounds, the slack tension. This rule is, however, influ¬ 
enced by the grade of elasticity of the belt. 

Example 10. How much tension must be given to a belt when at rest, in 
order to transmit when in motion a motive force F=500 pounds, when the 
angle of contact is 165°, or Z— 82° 30', and the friction coefficient/= 0.4? 


T+t == 


180X 500 
0.4 X 3.1416 X 82.5 


868.1 pounds. 


■The tension of the belt should be 868: 2 = 434 pounds, to which must be 
added yd for practical working, making the required tension 579 pounds. 
The friction coefficient is found by formulas 13 and 26. 

The formulas will answer equally well for any system of weights and 
measures. 










Transmission op Power by Belting. 


373 


Formulas for Oblique Belting, 

Figs. 1 and 2. 


(T+t) : W—l: sin. Z. 


Sum of tensions (T + t) — 


• W 


. . 1 . 


sin. Z 

Weight TF=(2 7 + 0sin. Z. .2 

TF 


Half-angl. cont. sin. Z= 


'( T+ty 


3. 


WnZ 

Force of contact C= — _ —- . . 4. 

180° sin. Z 

Force of contact (7= . . 5. 

loO 

180(7 

Sum of tensions (T + t)= O' . . . 6. 

IZi 

180°^ 

Half-ang. cont. Z= , m . ~ .7. 


«(T+i) 


_ T . . , , TIT 180°C'sin.^' „ 

Weight suspended TF=-—-. 8. 

TT A 

Motive force . 9 - 

180° F 

Sum of tensions (T +1) =*=———■. 10. 

j-aZ 

Greatest tension 

\ 180 —firZ ] 

Slack tensi0D<_r(|“^|). 12. 
Friction coett/- ... 13. 


Formulas for Parallel Belting 
on Pulleys of Equal Diam¬ 
eters. 


Sum of tensions (T + t) = W . ... 14. 

Pressure in journals W=(T + t). . 15. 

W 

Half-angl. cont. sin.^T=———=1 16. 

(2 + 1 ) 


TGr 

2 * * * 

n(T + t) 
2 

20 

Sum of tensions (T + t) = —. 


Force of contact C= 

Force of contact C‘ 


Half-angle cont. Z=90°. 


2(7 

Pressure in journals W ——. 

7T 


Motive force F= 


MT+t ) 


Sum of tensions (T+ t) = 


2 F 


Greatest tension T=t (^~^j 
Slack tension t =T ^ • • 


Friction coeff./= 


2 (T—t) 
"<T+t) 


. 17. 

. 18. 

. 19. 


. 20 . 

. 21 . 

. 22 . 

. 23. 

. 24. 

. 25. 


26. 


Friction Coefficient for Different Surfaces In Contact 


Surface 

of 

pulley. 

t 

: 

Cc 

Hair side 
ou pulley. 

ndition of 

Flesh 

side 

on pulley. 

leather be 

Wet belt. 

it. 

Good 

adhesive. 

India- 

rubber 

belt. 

Canvas 

belt. 

Gutta¬ 

percha 

belt. 


/ 

/ 

/ 

/ 

/ 

/ 

/ 

Rubber. 

0.50 

0.46 

0 42 


0 43 

0 30 

0 4*> 

Leather. 

0.48 

0.45 

0.50 

0.60 

0.42 

0^27 

0.40 

Wood. 

0.46 

0.40 

0.48 

0.55 

0.41 

0.23 

0.38 

Iron... 

0.40 

0.35 

0.45 

0.50 

0.38 

0.20 

0.35 















































374 


Transmission of Power by Belting. 


TABLE I.- 

-Motive Force P, when the Pulling Tension T = 1. 

Angle of 

Friction coefficient/, for the surfaces in contact on the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

‘IZ 

F 

F 

F 

F 

F 

F 

F 

F 

F 

F 

60° 

0.147 

0.189 

0.231 

0.272 

0.310 

0.346 

0.367 

0.415 

0.447 

0.462 

70° 

0.171 

0.219 

0.269 

0.316 

0.352 

0.393 

0.455 

0.467 

0.481 

0.536 

80° 

0.189 

0.245 

0.297 

0.346 

0.393 

0.436 

0.478 

0.514 

0.555 

0.598 

90° 

0.202 

0.274 

0.330 

0.388 

0.431 

0.477 

0.522 

0.544 

0.620 

0.648 

100 ° 

0.231 

0.300 

0.358 

0.415 

0.468 

0.517 

0.564 

0.608 

0.648 

0.687 

110 ° 

0.254 

0.325 

0.392 

0.453 

0.503 

0,542 

0.602 

0.648 

0.688 

0.731 

120 ° 

0.272 

0.346 

0.416 

0.478 

0.536 

0.590 

0.640 

0.687 

0.731 

0.793 

130° 

0.292 

0.373 

0.445 

0.515 

0.568 

0.624 

0.676 

0.724 

0.768 

0.810 

140° 

0.310 

0.393 

0.468 

0.536 

0.594 

0.656 

0.709 

0.758 

0.797 

0.846 

150° 

0.330 

0.418 

0.498 

0.570 

0.628 

0.687 

0.741 

0.791 

0.837 

0.880 

160° 

0.346 

0.436 

0.517 

0.591 

0.656 

0.717 

0.793 

0.822 

0.869 

0.912 

170° 

0.365 

0.461 

0.545 

0.623 

0.683 

0.745 

0.801 

0.852 

0.898 

0.942 

180° 

0.380 

0.478 

0.564 

0.640 

0.709 

0.772 

0.828 

0.880 

0.927 

0.970 

190° 

0.399 

0.499 

0.592 

0.671 

0.727 

0.797 

0.854 

0.906 

0.953 

0.997 

200 ° 

0.415 

0.517 

0.607 

0.687 

0.758 

0.822 

0.880 

0.932 

0.975 

1.000 

210 ° 

0.433 

0.539 

0.633 

0.717 

0.781 

0.846 

0.904 

0.956 

1.000 

1.000 

220 ° 

0.447 

0.555 

0.668 

0.731 

0.803 

0.868 

0.926 

0.979 

1.000 

1.000 

230° 

0.464 

0.571 

0.674 

0.758 

0.825 

0.890 

0.949 

1.000 

1.000 

1.000 

240° 

0.478 

0.590 

0.687 

0.772 

0.845 

0.912 

0.970 

1.000 

1.000 

1.000 

250° 

0.492 

0.612 

0.706 

0.795 

0.866 

0.932 

0.991 

1.000 

1.000 

1.000 

TABLE II.- 

—Pulling Tension T, 

when the 

Motive Force F = l. 

Angle of 


Friction coefficient/ for the surfaces in contact on the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 Z 

T 

T 

T 

T 

T 

T 

T 

T 

T 

T 

60° 

6.779 

5.291 

4.321 

3.680 

3.227 

2.887 

2.724 

2.410 

2.236 

2.163 

70° 

5.855 

4.558 

3.711 

3.165 

2.838 

2.546 

2.198 

2.140 

2.078 

1.863 

80° 

5.274 

4.077 

3.363 

2.887 

2.546 

2.290 

2.092 

1.944 

1.802 

1.671 

90° 

4.703 

3.646 

3.028 

2.580 

2.319 

2.092 

1.915 

1.838 

1.613 

1.543 

100 ° 

4.321 

3.364 

2.792 

2.410 

2.136 

1.933 

1.773 

1.645 

1.542 

1.455 

110 ° 

3.941 

3.079 

2.548 

2.206 

1.988 

1.845 

1.660 

1.542 

1.453 

1.368 

120 ° 

3.680 

2.887 

2.403 

2.091 

1.864 

1.693 

1.561 

1.455 

1.368 

1.261 

130° 

3.421 

2.683 

2.246 

1.942 

1.759 

1.601 

1.479 

1.381 

1.302 

1.234 

140° 

3.227 

2.546 

2.136 

1.864 

1.699 

1.523 

1.411 

1.319 

1.254 

1.182 

150° 

3.032 

2.390 

2.009 

1.755 

1.591 

1.455 

1.349 

1.264 

1.195 

1.137 

160° 

2.887 

2.290 

1.932 

1.690 

1.523 

1.395 

1.261 

1.216 

1.151 

1.097 

170° 

2.739 

2.169 

1.835 

1.581 

1.463 

1.342 

1.249 

1.174 

1.113 

1.062 

180° 

2.631 

2.091 

1.773 

1.561 

1.410 

1.296 

1.207 

1.136 

1.079 

1.030 

190° 

2.506 

2.004 

1.688 

1.490 

1.374 

1.253 

1.170 

1.103 

1.049 

1.003 

200 ° 

2.410 

1.932 

1.646 

1.455 

1.319 

1.216 

1.136 

1.073 

1.026 

1.000 

210 ° 

2.307 

1.853 

1.580 

1.395 

1.280 

1.182 

1.106 

1.046 

1.000 

1.000 

220 ° 

2.236 

1.802 

1.495 

1.368 

1.244 

1.151 

1.080 

1.021 

1.000 

1.000 

230° 

2.153 

1.730 

1.489 

1.318 

1.212 

1.123 

1.053 

1.000 

1.000 

1.000 

240° 

2.091 

1.693 

1.455 

1.296 

1.196 

1.098 

1.030 

1.000 

1.000 

1.000 

250° 

2.023 

1.633 

1.416 

1.257 

1.155 

1.073 

1.009 

1.000 

1.000 

1.000 

It is assumed in the above tables that the friction 

gripe on the smallest 

pulley just balances the motive force, for which allowance must be made to 

prevent slipping. 









The slack tension 2 

is the difference between the pulling tension Tand the 

motive force F, or f = 

= T — F. 







When the friction 

and angle of contact are great, the pulling tension is 

equal to the motive force, 

and no slack tension is then required. 











































Transmission of Power. 


375 


TABLE III.—Pressure P in tile Shaft Journals, when the 31 o- 
tive Force F = 1 anti the System in Motion. 


Angle of 

Friction coefficient / for the surfaces in contact on the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 Z 

P 

P 

P 

P 

P 

P 

P 

P 

P 

p 

60° 

6.779 

5.291 

4325 

3.680 

3.227 

2.887 

2.724 

2.410 

2.236 

2.163 

70° 

6.570 

5.082 

4.110 

3.484 

3.109 

2.774 

2.414 

2.308 

2.135 

1.989 

80° 

6.495 

4.956 

4.038 

3.352 

2.988 

2.658 

2.339 

2.214 

2.030 

1.863 

90° 

6.237 

4.742 

3.868 

3.235 

2.865 

2.544 

2.294 

2.093 

1.867 

1.768 

100 ° 

6.088 

4.624 

3.746 

3.100 

2.740 

2.430 

2.184 

1.988 

1.829 

1.697 

110 ° 

5.818 

4.406 

3.536 

2.976 

2.619 

2.384 

2.081 

1.888 

1.701 

1.602 

120 ° 

5.642 

4.268 

3.430 

2.890 

2.497 

2.200 

1.972 

1.788 

1.637 

1.451 

130° 

5.388 

4.051 

3.259 

2.708 

2.376 

2.089 

1.868 

1.691 

1.547 

1.424 

140° 

5.185 

3.906 

3.135 

2.624 

2.257 

1.983 

1.772 

1.600 

1.477 

1.342 

150° 

4.925 

3.685 

2.949 

2.456 

2.142 

1.879 

1.674 

1.510 

1.377 

1.264 

160° 

4.717 

3.541 

2.836 

2.359 

2.030 

1.778 

1.513 

1.431 

1.297 

1.191 

170° 

4.465 

3.329 

2.664 

2.157 

1.922 

1.681 

1.496 

1.351 

1.224 

1.122 

180° 

4.262 

3.182 

2.546 

2.122 

1.824 

1.592 

1.414 

1.272 

1.158 

1.060 

190° 

4.000 

3.000 

2.371 

1.976 

1.745 

1.504 

1.339 

1.169 

1.097 

1.005 

200 ° 

3.777 

2.836 

2.261 

1.896 

1.628 

1.425 

1.268 

1.134 

1.050 

1.000 

210 ° 

3.525 

2.648 

2.120 

1.763 

1.541 

1.352 

1.205 

1.101 

1.000 

1.000 

220 ° 

3.323 

2.507 

1.930 

1.692 

1.458 

1.284 

1.150 

1.020 

1.000 

1.000 

230° 

3.090 

2.323 

1.886 

1.576 

1.384 

1.223 

1.066 

1.000 

1.000 

1.000 

240° 

2.890 

2.200 

1.788 

1.513 

1.340 

1.170 

1.032 

1.000 

1.000 

1.000 

250° 

2.676 

2.037 

1.681 

1.421 

1.254 

1.120 

1.015 

1.000 

1.000 

1.000 

TABLE IV. 

—Pressure P in the Shaft Journals, when the Mo- 


tive Force F = X and the System at Rest. 


Angle of 

Friction coefficient f for the surfaces in contact on 

the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2Z 

P 

P 

P 

P 

P 

P ■ 

P 

P 

P 

P 

60° 

5.779 

4.291 

3.325 

2.680 

2.227 

1.887 

1.724 

1.410 

1.236 

1.163 

70° 

5.570 

4.082 

3.110 

2.484 

2.109 

1.774 

1.444 

1.308 

1.135 

0.989 

80° 

5.495 

3.956 

3.038 

2.352 

1.988 

1.658 

1.339 

1.214 

1.030 

0.862 

90° 

5.237 

3.742 

2.863. 

2.235 

1.865 

1.544 

1.294 

1.093 

0.867 

0.768 

100 ° 

5.088 

3.624 

2.746 

2.100 

1.740 

1.430 

1.184 

0.988 

0.829 

0.697 

110 ° 

4.818 

3.406 

2.536 

1.976 

1.619 

1.384 

1.081 

0.888 

0.701 

0.602 

120 ° 

4.642 

3.268 

2.430 

1.890 

1.497 

1.200 

0.972 

0.788 

0.637 

0.451 

130° 

4.388 

3.051 

2.259 

1.708 

1.376 

1.089 

0.868 

0.691 

•0.547 

0.424 

140° 

4.185 

2.906 

2.135 

1.624 

1.257 

0.983 

0.772 

0.600 

0.477 

0.342 

150° 

3.925 

2.685 

1.949 

1.456 

1.142 

0.879 

0.674 

0.510 

0.377 

0.264 

160° 

3.717 

2.541 

1.836 

1.359 

1.030 

0.778 

0.513 

0.431 

0.297 

0.191 

170° 

3.465 

2.329 

1.664 

1.157 

0.922 

0.681 

0.496 

0.351 

0.224 

0.122 

180° 

3.262 

2.182 

1.546 

1.122 

0.824 

0.592 

0.414 

0.272 

0.158 

0.060 

190° 

3.000 

2.000 

1.371 

0.976 

0.745 

0.504 

0.339 

0.169 

0.097 

0.005 

200 ° 

2.777 

1.836 

1.261 

0.896 

0.628 

0.425 

0.268 

0.134 

0.050 

0.000 

210 ° 

2.525 

1.648 

0.120 

0.763 

0.541 

0.352 

0.205 

0.101 

0.000 

0.000 

220 ° 

2.323 

1.507 

0.930 

0.692 

0.458 

0.284 

0.150 

0.020 

0.000 

0.000 

230° 

2.090 

1.323 

0.886 

0.576 

0.384 

0.223 

0.066 

0.000 

0.000 

0.000 

240° 

1.890 

1.200 

0.788 

0.513 

0.340 

0.170 

0.032 

0.000 

0.000 

0.000 

250° 

1.676 

1.037 

0.681 

0.421 

0.254 

0.120 

0.015 

0.000 

0.000 

0.000 

The belt should be tightened to the pressure pin the journals when the 

svstein is at rest, to 

enable it to transmit the 

motive force i^when the sys- 

tern is in motion. The friction gripe on the 

smallest pulley will then just 

balance the motive force, or C=F=P— 

-p. The pressure p should therefore 

be made | greater for safe working. 














































>76 


Transmission of Power by Belts. 


CONE PULLEYS. 

The illustration Fig. 2, Plate III., represents the largest and smallest diam¬ 
eters of a pair of cone pulleys, of which letters denote as follows: 

R = radius of the largest step or pulley. 
r = radius of the smallest step or pulley. 
a = distance between the centres of rotation of the pulleys. 

& = distance or length of the belt between the tangenting points on the 
two pulleys. 

L= whole length of the belt. 

£ = distance from the centre of the small pulley to the vertex c f . 

Z = half the angle of contact of the belt on the large pulley, 
if = half the angle of contact of the belt on the small pulley. 

For open belts the sum of the whole angles of contact is always 360°, and 
Z + Z= 180°, omitting the slack of belt. 

—••••*. 

L = ~^{RZ + rZ)-f2b . . 4. 

Tliree-step Pulley, Fig. 3, Plate III, 

The object is to make two cone pulleys cast from one pattern, and to have 
three steps on each pulley. Having given the diameters of the largest and 
smallest steps, the problem is to find the diameter of the middle step, so pro¬ 
portioned that the belt will have the same tension on all the three steps. 
i> = diameter of the middle step. 

D = R + r + .5. 

TTCl 

Pour-step Pulley, Fig. 4, Plate III. 

Having given the radii of the largest and smallest pulleys 1 and 4, the 
problem is to find the diameters of the inner pulleys 2 and 3. 
d = diameter of the pulley 3, next to the smallest. 
d'= diameter of the pulley 2, next to the largest pulley. 

<1 = 3 (R + 2r) + - ....... 6. 

d'= | (2R + r) + .7. 


& = |/ a 2 — (/f — r)2 . . . . 1, 

Sin. Z= — .2. 

a 


Five-step Pulley, Fig. 5, Plate III. 

Cone pulleys of five steps are constructed as follows: The largest and 
smallest pulleys, 1 and 5, are assumed to be given from the first start. The 
diameter of the middle step 3 is calculated by formula 5, and the problem 
remains to find the diameters of the steps 2 and 4. 

i2' = radius of the middle step 3; d and d' = diameters of the respective 
steps 4 and 2. 

d=R' + r + ( R '~ r ' )2 .8. 

ira 

d f = R+PJ+ .9. 

Tva 

It is supposed in the above that each pair of cone pulleys is cast from one 
pattern. 

To And tlie Proper Distance between Centres. 


Having given two equal cone pulleys, to find at what distance they ought 
to be placed to make the belt of equal tension on all the steps. 

(R— r)* 

a = -\. 10 . 

it (D — R — r) 

This formula can be used only for pulleys of an odd number of steps, of 
which D= diameter of the middle step, and R and r are the respective radii 
of the largest and smallest pulleys. 






















Transmission op Power. 


377 


Conte Pulleys of any Number of Steps and Proportions, Figs. 

6 and 7, Plate III. 

The conoids a, b, c d B and Ad cb a are drawn alike and inverted to one 
another, as represented by Fig. 6. The centre lines A B can be divided into 
any number of steps, and the conoids determine the corresponding diameters 
on each pulley. The pulleys can thus be made of widely different sizes, as 
required in many cases, particularly in foot-lathes. 

The construction of the conoid is represented by Fig. 7. Draw the centre¬ 
line A B, in the middle of which erect the mean diameter _D, which will be 
alike on each pulley. The largest radius R is determined by the formula 


, na 
+ naD — —. 


11 . 


Divide the centre line A B into four equal parts and draw the diameters d 
and d', the lengths of which are determined by the following formulas : 


d = iD + 


ira ' 


12 . 


d' — R -j- 3D -f- 


(R-m°- 


ira 


13. 


Draw a regular curve through the points a, b, c , d and B which form the 
sides of the conoid. Draw also the inverted conoid as shown by the dotted 
lines. Any line drawn through the conoids at right angles to A B determines 
t he diameters of the corresponding steps on each cone pulley. The diameter 
b o on one pulley corresponds to the diameter m n on the other pulley. It is 
not intended that the conoid should determine the width of the steps, as may 
be inferred from Fig. 6, for it only determines the diameters of the steps. 

In practice the conoids should be laid out on a full-size scale to make the 
measurements correct. The length of the centre-line A B should not be less 
than the diameter D , but better to make it 2 D, for the less inclination of the 
conoid makes sharper measurements for the diameters of the steps. 

The whole length of the belt will be, 

L =?= itD -f 2a. . .14. 

For different lengths of the belt and diameter D, with the same distance a 
between the shafts, the conoid a, 6, c, d, B will be of more or less curvature. 
When a =co, the conoid becomes a straight line, and when a = B, the radius 
R = 0.8B, which forms the greatest curvature of the conoid. 


For a given length of belt the diameter of the middle step should be 

L — 2 a 


D 


15. 


The diameter of the middle step can also be obtained by assuming a def¬ 
inite value on the greatest radius R —namely : 


R? 

D = i2 + —. 

jrn 


The length of the belt will then be— 


R 2 

B — ttR -J- " ■ -|- 2 a, 
a 


. 16. 


17. 


Having given the greatest radius R and middle diameter D, the distance 
between the shafts should be— 

R* 

a t t(D-R)' ’ * .. 













378 


Power op Belting. 


Dimensions, Strength, and Power of Belts. 

The strain on a belt is its pulling tension T, and not ouly the motive force 
I F, as is often considered. 

i The motive force, under some conditions, is only a small fraction of the 
pulling tension, as seen in the Tables I. and II., page 374. 

The strength of the belt must, therefore, be in proportion to the pulling 
tension T. 

The following formula 1 is more correct for calculating the breadth of belts 
than the formulas on page 265. The difference is, however, very small. 

S = maximum strain in pounds per inch of width of belt, which should not 
exceed the safety strength given in the accompanying tables, but may 
be made less. 

B — breadth of belt in inches. T= pulling tension in pounds. 

Breadth of belt B — ..1. 

/O 

Pulling tension T= BS. .2. 

T 

Strain per inch S = —.3. 

* X> 

India-rubber belts are best in wet or damp places where leather belts can¬ 
not be used. 


Dimensions and Strength of India-rutober Belts. 


Number of plies. 

Wt. per. sq. ft. 
Pounds. 

Thickness. 

Inches. 

TJlt. strength. 
Lbs. per in. 

Safety 

strength. 

2 ply. 

1.25 

T 3 H = 0.1875 

625 

104 

3 ply. 

1.66 

& = 0.2083 

830 

138 

4 ply. 

2 

= 0.3125 

1000 

166 

5 ply. 

2.4 

— 0.4166 

1200 

200 

6 ply. 

2.8125 

X 7 S = 0.4375 

1400 

233 


Thickness in Inches and Strength in Pounds of Belts. 


Kind of material in belts. 

Thickness. 

Stre 

Break. 

ugth. 

Safety. 

Oak-tanned leather. 

0.25 

1000 

166 

U U 

0.1875 

780 

130 

u u 

0.125 

560 

95 

Ordinary tanned leather. 

0.25 

740 

125 

U <( U 

0.1875 

560 

95 

u u u 

0.125 

290 

50 

Raw hide, best quality. . 


1250 

225 

“ “ ordinary.. 


1100 

185 

Horse-skin. 


800 

135 

Calf’s-skin. 


360 

60 

Sheep-skin. 


322 

54 

Cowhide. 


790 

130- 

Cotton duck. 


200 

66 

Flax, woven belt. 


1250 

200 


The above data are for new belts, and cannot be trusted for old and worn- 
out belts. 


Care should be taken to prevent animal oil or fat from coming in contact 
with the working surfaces of the belt and pulleys, for it reduces the friction, 
and if once got into the leather it is difficult to get rid of. 


















































T rcurusmu s s i on of Poi ue r. Plate HI . 











































































































" 

























* 



- 

























- 




















• 















































Heat. Caloric. 


379 


HEAT. CALORIC. 


The physical constitution of heat is yet under investigation by operative minds : 
its well-known character and effect upon matter is the base for the investigation. 

Heat resembles light, electricity and magnetism. It is convertible into dynamic 
work, and can consequently be resolved into the three physical elements, force , 
velocity and time. Temperature is convertible into force, which is only one ele¬ 
ment of heat, and is no measure of quantity of heat. (See Dynamics and Units 
of Heat.) 

The temperature or intensity of heat is measured in various ways, but most 
generally by the expansion of mercury and alcohol, or the thermometer. 


Thermometers. 

There are three differently-graduated thermometers in use—namely, Fahrenheit, 
Centigrade, and Reaumur. The last named is gradually being abolished, and now 
used only in Peru. 

Graduation. Fahr. Cent. Keau. 


Zero Fahr. = —17.77° Cent—14.22° Reau. 

Freezing-Point of Water. 

Zero Cent. = 32 Fahr. = zero Reau. 
Boiling-Point of Water. 

212° Fahr. = 100° Cent. = 80° Reau. 

9° Fahr. = 5° Cent. = 4° Reau. 

Formulas. 

Cent. = f- (Fahr. hF 32) = f- Ream 
Fahr. = f Cent. ± 32 = f Ream ± 32. 
Reau. — f Cent. =’ y (Fahr. =F 32). 



The accompanying tables give the equivalents of Centigrade’s and Fahrenheit's 
thermometers. The tirst numbers in the table of comparison, — 276* and 461*, are 
the absolute zero of temperature. 

Example. 1. How many degrees on Fahr. scale is 964.5° Cent. ? 

Table comparison, Cent. 960° — 1760° Fahr. 

Table Centigrade, Cent. 4.5 = 8.1 “ 

The required, ~ Cent. 964.5 1768.1 ft 

Example 2. How many degrees is 2136.7° Fahr. on Centigrade thermometer? 

Table comparison, Fahr. 2120° = 1160° Cent. 

Table Fahrenheit, “ 16° = 8.90 “ 

“ “ “ 0.7 = 0.389 “ 

The required degrees, Fahr. 2136.7 — 1169.289 “ 

















380 


Thermometers. 


Comparison of Fahrenheit and Centigrade Thermometers. 


Falir. 

Centig. 

Falir. 

Centig. 

Fahr. 

Centig. 

Falir. 

Crntig. 

Fahr. 

Centig. 

— 6 

— 20.55 

57 

13.88 

119 

48.33 

181 

82.77 

243 

117.22 

— 4 

— 20.00 

58 

14.44 

120 

48.88 

182 

83.33 

214 

117.77 

— 3 

— 19.44 

69 

15.00 

121 

49.44 

183 

83.88 

245 

118.33 

— 2 

— 18.88 

60 

18.55 

122 

50.00 

184 

84 44 

246 

118.88 

— 1 

—18.33 

61 

16.11 

123 

50.55 

185 

85.00 

247 

119.44 

Zero. 

—17.77 

62 

16.66 

124 

51.11 

186 

85.55 

248 

120.00 

+ 1 

— 47.22 

63 

17.22 

125 

51.66 

187 

86.11 

249 

120.55 

2 

—16.66 

64 

17.77 

126 

52.22 

188 

86.66 

250 

121.11 

3 

—16.11 

65 

18.33 

127 

52.77 

189 

87.22 

251 

121.66 

4 

— 15.55 

66 

18.88 

128 

53.33 

190 

87.77 

252 

122.22 

5 

—15.00 

67 

19.44 

129 

53.88 

191 

88.33 

253 

122.77 

6 

—14.44 

68 

20.00 

130 

54.44 

192 

88.88 

254 

123.33 

7 

—13.88 

69 

20.55 

131 

55.00 

193 

89.44 

255 

123.88 

8 

—13.33 

70 

-1.11 

132 

55.55 

194 

90.00 

256 

124.44 

9 

—12.77 

71 

21.66 

133 

56.11 

195 

90.55 

257 

125.00 

10 

—12.22 

72 

22.22 

134 

56.66 

196 

91.11 

258 

125.55 

11 

— 11.66 

73 

22.77 

135 

57.22 

197 

91.66 

259 

126.11 

12 

— 11.11 

74 

23.33 

136 

57.77 

198 

92.22 

260 

126.66 

13 

—10.55 

75 

23 88 

137 

58.33 

199 

92.77 

261 

127.22 

14 

—10.00 

76 

24.44 

138 

58.88 

200 

93.33 

262 

127.77 

15 

— 9.44 

77 

25.00 

139 

59.44 

201 

93.88 

203 

128.33 

16 

— 8.88 

78 

25.55 

140 

60.00 

202 

94.44 

264 

128.88 

17 

— S 33 

79 

26.11 

141 

60.55 

203 

95.00 

265 

129.44 

18 

— 7.77 

80 

26.66 

142 

6i.ll 

204 

95.56 

266 

130.00 

19 

— 7.22 

81 

27.22 

143 

61.66 

205 

96.11 

267 

130.55 

20 

— 6.66 

82 

27.77 

144 

62.22 

206 

96.66 

268 

131.11 

21 

— 6.11 

83 

28.33 

145 

62.77 

207 

97.22 

269 

131.(6 

22 

— 5.55 

84 

28.88 

146 

63.33 

208 

97.77 

270 

132.22 

23 

— 5.00 

85 

29.44 

147 

63.88 

209 

98.33 

271 

132.77 

24 

— 4.44 

86 

30.00 

148 

64.44 

210 

98.88 

272 

133.33 

25 

— 3.88 

87 

30.55 

119 

65.00 

211 

99.44 

273 

133.88 

26 

— 3.33 

88 

31.11 

150 

65.55 

212 

100 00 

274 

134.44 

27 

— 2.77 

89 

31.66 

151 

(.6.11 

213 

100.55 

275 

135.00 

28 

— 2.22 

90 

32.22 

152 

66.66 

214 

101.11 

276 

135.55 

29 

— 1.66 

91 

32.77 

153 

67.22 

215 

101.66 

277 

136.11 

30 

— 1.11 

92 

33.33 

154 

67.77 

216 

102.22 

278 

136.66 

31 

— .55 

93 

33.88 

155 

68.33 

217 

102.77 

279 

137.22 

32 

Zero. 

94 

34.44 

156 

08.88 

218 

•103.33 

280 

137.77 

33 

+ 0.55 

95 

35.00 

157 

69.44 

219 

103.88 

281 

138.33 

34 

1.11 

96 

35.55 

158 

70.00 

220 

104.44 

282 

138.88 

35 

1.66 

97 

36.11 

159 

70.55 

221 

105.00 

283 

139.44 

36 

2.22 

98 

36.66 

160 

71.11 

222 

105.55 

284 

140.00 

37 

2.77 

99 

37 22 

161 

71.66 

223 

106.11 

285 

140.55 

38 

3.33 

100 

37.77 

162 

72.22 

224 

106.66 

286 

141.11 

39 

3.88 

101 

38.33 

163 

72.77 ■ 

225 

107.22 

287 

141.66 

40 

4.44 

102 

38.88 

164 

73.33 

226 

107.77 

288 

142.22 

41 

5.00 

103 

• 39.44 

165 

73.88 

227 

108.83 

289 

142.77 

42 

5.55 

104 

40.00 

166 

74.44 

228 

10S.88 

290 

143.33 

43 

6.11 

105 

40.55 

167 

75.00 

229 

109.44 

291 

143.88 

44 

6 66 

106 

41.11 

168 

75.55 

230 

110.00 

292 

144.44 

45 

7.22 

107 

41.66 

169 

76.11 

231 

110.55 

293 

145.00 

46 

7.77 

108 

42.22 

170 

76.66 

232 

111.11 

294 

145.55 

47 

8.33 

109 

42.77 

171 

77.22 

233 

111.66 

295 

146.11 

48 

8.88 

110 

43.33 

172 

77.77 

234 

112.22 

296 

146.66 

49 

9.44 

111 

43.38 

173 

78.33 

235 

112.77 

297 

147.22 

60 

10.00 

112 

44.44 

174 

78.88 

236 

113.33 

298 

147.7 7 

51 

10.55 

113 

45.00 

175 

79.44 

237 

113.88 

299 

148.33 

52 

11.11 

114 

45.55 

176 

80.00 

238 

114.44 

300 

148.88 

; 53 

11.66 

115 

46.11 

177 

80.55 

239 

116.00 

400 

204.44 

54 

12.22 

116 

46.66 

178 

81.11 

240 

115.55 

600 

315.55 

55 

12 77 ! 

117 

47.22 

179 

81.66 

211 

11 6.11 

800 

433.33 

66 

13.33 | 

118 

47.77 

180 

82.22 

242 

116.66 

1000 

537.77 




























Thermometers, 


383 


- 


' 








Comparison of Centigrade and Fahrenheit Thermometers. 

Cent. 

Fahr. 

Cent. 

Fahr. 

Cent. 

Fahr. 

Cent. 

Fahr. 

Cent. 

Fahr. 

276 s 

461* 

16 

60.8 

330 

626 

950 

1742 

157 o 

2858 

—260 

--436 

17 

62.6 

340 

644 

9 0 

J760 

1580 

2876 

—250 

—418 

18 

64.4 

350 

662 

970 

1778 

1590 

2894 

—240 

—400 

19 

66.2 

360 

680 

980 

1796 

1600 

2912 

—230 

—382 

20 

68.0 

370 

698 

990 

1814 

1610 

2930 

—220 

—364 

21 

69.8 

380 

716 

1000 

1832 

1620 

2948 

—210 

—346 

22 

71.6 

390 

734 

1010 

1850 

1630 

2966 

—200 

—328 

23 

73.4 

400 

752 

1020 

1868 

1640 

2934 

—190 

—310 

24 

75.2 

410 

770 

1030 

1886 

1650 

3002 

—180 

—298 

25 

77.0 

420 

788 

1040 

1901 

16 >0 

3020 

—170 

—274 

26 

78.8 

430 

806 

1050 

1922 

1670 

3038 

—160 

—256 

27 

80.6 

440 

824 

1060 

1940 

1680 

3056 

— 150 

—238 

28 

82.4 

450 

842 

1070 

1958 

1690 

3074 

—140 

—220 

29 

84.2 

460- 

860 

1080 

1976 

170J 

3092 

—130 

—202 

80 

86.0 

470 

878 • 

1090 

1994 

1710 

3110 

—120 

—184 

31 

87.8 

480 

896 

1100 

2012 

1720 

3128 

— TO 

—166 

32 

89.6 

490 

914 

1110 

2030 

1730 

3146 

—100 

—148 

33 

91.4 

500 

932 

1120 

2048 

1740 

3164 

— 90 

—130 

34 

93.2 

510 

950 

1130 

206 5 

1750 

3182 

— 80 

—112 

&5 

95.0 

520 

968 

1140 

2084 

1760 

3200 

— 70 

— 94 

36 

96.8 

530 

986 

1150 

2102 

1770 

3218 

— 60 

— 76 

37 

98.6 

540 

1004 

1160 

2120 

1780 

3236 

— 50 

— 58 

38 

100.4 

550 

1022 

1170 

2138 

1790 

3254 

— 40 

— 40 

39 

102.2 

560 

1040 

1180 

2156 

1800 

3272 

— 30 

— 22 

40 

104.0 

570 

1058 

1190 

2174 

1810 

3290 

— 20 

— 4 

41 

105.8 

580 

1076 

1200 

2192 

1820 

3308 

— 19 

— 2.2 

42 

107.6 

590 

1094 

1210 

2210 

1830 

3326 

— 18 

— 0.4 

43 

109.4 

600 

1112 

1220 

2228 

1840 

3344 

17.77 

Zero. 

44 

111.2 

610 

1130 

1230 

2246 

1850 

3362 

— 17 

+ 1-4 

45 

113.0 

620 

1148 

1240 

2264 

1360 

3380 

— 1G 

+ 3.2 

46 

114.8 

630 

1166 

1250 

2282 

1*70 

3398 

— 15 

+ 5.0 

47 

116.6 

640 

1184 

1260 

2300 

1880 

3416 

— 14 

+ 6.8 

48 

118.4 

650 

1202 

1270 

2318 

1*90 

3434 

— 13 

+ 8.6 

49 

120.2 

660 

1220 

1280 

2336 

1900 

3452 

— 12 

+10.4 

50 

122.0 

670 

1238 

1290 

2354 

1910 

3470 

— 11 

+12.2 

60 

140 

680 

1256 

1300 

2372 

1920 

3488 

— 10 

+ 14.0 

70 

158 

690 

1274 

1310 

2390 

1930 

3506 

— 9 

+15.8 

80 

176 

700 

1292 

1320 

2408 

1940 

3524 

— 8 

+ 17.6 

90 

194 

710 

131 ) 

133 ) 

2426 

1950 

3542 

— 7 

+19.4 

100 

212 

720 

1328 

1340 

2414 

1960 

3560 

— 6 

+21.2 

110 

230 

730 

1346 

1350 

2462 

1970 

{8 

— 5 

+23.0 

120 

248 

740 

1364 

1360 

2480 

1980 

3596 

— 4 

+24.8 

130 

266 

750 

1382 

1370 

2498 

1990 

3614 

— 3 

+26.6 

140 

28 4 

760 

1400 

1380 

2516 

2000 

3632 

— 2 

+28.4 

150 

302 

770 

1418 

1390 

2534 

2010 

3650 

— 1 

+30.2 

160 

320 

780 

1436 

1400 

2552 

2020 

3668 

Zcr .. 

+32. 

170 

338 

7.)0 

1454 

1410 

' 2570 

2030 

3686 

+1 

+33.8 

180 

356 

800 

1472 

1420 

■ 2583 

2 140 

3704 

2 

35.6 

190 

374 

810 

1490 

1430' 

2606 

2050 

3722 

3 

37.4 

200 

392 

820 

1508 

1440 

2624 

2060 

3740 

4 

39.2 

210 

410 

830 

1526 

1450 

2642 

2070 

3758 

5 

41.0 1 

220 

428 

840 

1514 

1400 

2660 

2080 

3776 

6 

42.8 I 

230 

446 

850 

1562 

1470 

2678 

2090 

3794 

7 

44.6 1 

240 

464 

860 

1580 

1480 

2696 

2100 

38t2 

8 

46.4 

250 

482 

870 

1698 

1490 

2714 

2110 

3830 

9 

48.2 1 

260 

500 

880 

1616 

1500 

2732 

2120 

3848 

10 

50.0 

270 

518 ‘ 

890 

1634 

1510 

2750 

2130 

4166 

11 

51.8 

280 

536 

900 

1652 

1520 

2768 

2140 

4184 

12 

53.6 

290 

55 4 

910 

1670 

1730 

2786 

2150 

4162 

13 

52.4 

300 

572 

920 

1688 

1540 

2804 

2:60 

4180 

14 

57.2 

310 

590 

930 

1706 

1550 

2822 

200 

4216 

15 

59.0 

320 

608 

940 

1724 

1560 1 

2840 1 

2260 

4252 

-—J 




































3S2 


Comparison op Thermometers. 


Number of Degrees Cent. = Number of Degrees Falir. 


Degrees 


Tenths of a Degree—Centigrade Scale. 


Cent. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

0 

0.(10 

0.18 

0.36 

0.54 

0.72 

0.90 

1.08 

1.26 

1.44 

1.62 

1 

l.cSO 

1.98 

2.16 

2.34 

2.55 

2.70 

2.88 

3.06 

3.24 

3.42 

2 

3.60 

3.78 

3.96 

4.14 

4.32 

4.50 

4.68 

4.86 

5.04 

5.22 

3 

5.4i) 

5.58 

5.76 

5.91 

6.12 

6.30 

6.18 

6.66 

6.84 

7.02 

4 

7.20 

7.38 

7.56 

7.74 

7.92 

8.10 

8.28 

8.46 

8.64 

8.82 

5 

9.00 

9.18 

9.36 

9.54 

9.72 

9.90 

10.08 

10.26 

10.44 

10.62 

6 

10.80 

.10.98 

11.16 

11.34 

11.52 

11.70 

11.88 

12.06 

12.21 

12.42 

7 

12.GO 

12.78 

12 96 

13.14 

13.32 

13.50 

13.68 

13.86 

14.01 

14.22 

8 

14.40 

14.58 

14.76 

14.91 

15.12 

15.30 

15.48 

15.66 

15.84 

16.02 

9 

16.20 

16.38 

16.56 

16.74 

16.92 

17.10 

17.28 

17.46 

17.61 

17.82 


Number of Degrees Falir. = Number of Degrees Cent. 


Degrees 


Tenths of a Degree—Fahrenheit Scale. 


Fahr. 

.0 

.1 

.2 

• .3 

.4 

•5 

.6 

7. 

.8 

.9 


Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent 

0 

0.00 

0.06 

0.11 

0.17 

0.22 

0.28 

0.33 

0.39 

0.41 

<).5'> 

1 

0.56 

0.61 

0.67 

0.72 

0.78 

0.83 

0.89 

6.94 

1.00 

1.06 

2 

1.11 

1.17 

1.22 

1.28 

1.33 

1.39 

1.44 

1.50 

1.56 

1 .6i 

3 

1.67 

1.72 

1.78 

1.83 

1.89 

1 94 

2.00 

2.06 

2.11 

2., 7 

4 

2.22 

2 .2S 

2 33 

2.39 

2.44 

2.50 

2.56 

2.61 

2.67 

2 7_ 

5 

2.78 

2 83 

2.S9 

2.94 

3.00 

3.06 

3.11 

3.17 

3.22 

3.28 

6 

3.33 

3.39 

3.44 

3.50 

3.56 

3.61 

3.67 

3.72 

3.7 S 

3.8 

7 

3 89 

3.91 

4.00 

4.06 

4.11 

4.17 

4.22 

4.28 

4.33 

4.39 

8 

4.44 

4.50 

4 56 

4.61 

4 67 

4.72 

4.78 

4.83 

4.S ) 

4.94 

9 

5.00 

5.06 

5.11 

5.17 

5.22 | 

5.28 

5 33 

5.39 | 

5.14 

5.50 


LATENT HEAT. 


Latent heat is the number of units of heat required to change the aggregate 
form of a body, whilst the temperature remains constant—that is, tire heat required 
to melt a body from solid to liquid, and to evaporate a liquid. In the one case it i.- 
called the latent heat of fusion, and in the other, the latent heat of evaporation. 

liatent Units of Heat per Pound of Substance. 


Solids smelted to 
liquid. / 

Latent 

heat. 

Liquids converted to 
vnpor. 

Latent 

heat. 

Ice to water,. 

141 

Water to steam, .... 

966 

Tin,. 

25.6 

Ammonia,. 

895 

Zinc,. 

50.6 

Alcohol, pure,. 

372 

Sulphur,. 

17.0 

Carbonic acid, .... 

298 

Lead, . 

9.72 

Bisulphide of carbon, . . 

212 

Mercury,. 

5.00 

Ether, sulphuric, . . . 

174 

Beeswax, . 

175 

Essence of turpentine, . . 

137 

Bismuth,. 

550 

Oil of turpentine, . „ . 

181 

Cast iron,. 

233 

Mercury, ....... 

157 

Spermaceti,. 

46.4 

Chyiuogene,. 

175 

Fusion. 

Evaporation. 


L — latent heat (units) per pound of 
liquid at smelting-point. 

(7= specific heat of the liquid. 
c = specific heat of solid. 
t = temperature of fusion, Fahr. 

L = (C—c)(t+ 256). 


I — latent units of heat per pound of 
vapor at boiling-point. 

T— temperature of boiling-point, Fahr. 

( Regnault .) 

I = 1091.7 — 0.695( T — 32)— 
0.000000103(.T—39.1 ) 3 . 


















































































Temperature of Boiling and Evaporation. 


3SJ 


Temperature of Boiling or Evaporation under Atmospheric 

Pressure. 


Liquids. 

Fahr. 

Cent. 

Liquids. 

Fahr. 

Cent. 

Wrought iron, 

5000° 

2760° 

Alcohol, .... 

173 

78 

Cast iron, 

3300 

1815 

Ether, .... 

96 

35 

Mercury, . . 

665 

352 

Carbon, bisulphuretted, 

116 

47 

Whale oil, . 

630 

332 

Water, distilled, . 

212 

100 

Oil of linseed, . 

600 

316 

Salt sea water, 

213 

101 

Oil of turpentine, . 

357 

180 

Water 20 per cent, salt, 

218 

103 

Sulphuric acid, 

£93 

312 

“ 30 “ “ . 

222 

ro5 

Sulphur, 

Phosphorus, . 

570 

300 

“ 40 “ saturated, 

227 

108 

557 

292 

Ammonia, liquid , . 

140 

60 

Sweet oil, . 

412 

211 

Water in vacuo, . 

98 

36 

Naphtha, 

320 

160 

Chymogene, . 

-f 38 

3.3 

Nitric acid, . 

220 

104 

Carbonic acid, 

— 112 

— 80 

Milk of cows, . 

Rectified petroleum, . 

213 

101 

Ammonia, 

— 30 

— 34 

316 

158 





Distillation Temperatures of Coal-oils.— (Tissandier.) 


Light Oii.s. 

Fahr. 

Cent. 

Heavy Oils. 

Fahr. 

Cent. 

Heavy Oils. 

Fahr. 

Cent. 

Amvlene, 

102° 

38.9° 

Cumene, 

304° 

151° 

Carbolic acid 

370° 

188° 

Benzine, 

187° 

86.1° 

Lutidine, 

311° 

155° 

Nephthline, 

422° 

217° 

Toluene, 

226° 

108° 

Eupione, 

338° 

170° 

Quiloneine, 

462° 

239° 

Xylene, 

271° 

133° 

Cymene, 

347° 

175° 

Anthracene, 

500° 

260° 

Pyridine, 

302° 

150° 

.Aniline, 

359° 

182° 

Chrysene, 

572° 

300° 


The temperature of distillation of vapors is equal to that of the boiling-point of 
the liquid of which the vapor is formed. . 


Temperature of Fusion, Freezing or Smelting-Point. 


Solids. 



Fahr. 

Cent. 

Solids. 

Fahr. 

Cent. 

Platinum, 



3080° 

1693° 

Puddle slag, . . 

2606° 

1430° 

Wrought iron, 



2912 

1600 

Sulphur, . . . 

228 

109 

Cast iron, gray. 

• 


2012 

1100 

Beeswax, white, 
yellow, 

155 

68 

Cast iron, white, . 



1922 

1050 

142 

61 

Steel, 

• 


2500 

1371 

Spermaceti, 

142 

61 

Gold, pure, . 



2300 

1260 

Potassium, . 

186 

58 

Gold, money, . 

• 


2192 

1200 

Sodium, .... 

104 

90 

Copper, 



2160 

1232 

Olive oil, 

92 

33 

Biass, common, 



1900 

1038 

Tallow, .... 

36 

2.2 

Silver, . . . 



1850 

1021 

Ice of water, 

32 

0.600 

Litharge, 



1739 

954 

“ milk, . 

30 

— 1.1 

Antimony, . 



800 

427 

“ sea water,. . 

28 

— 2.2 

Zinc, 



740 

393 

“ vinegar, 

28 

— 2.2 

Lead, . 


. L 

600 

316 

“ strong wine, 

20 

— 6.6 

Bismuth,. 


B 

470 

254 

“ “ brandy, 

7 

— 13.9 

Tin, . 


. T 

420 

215 

“ oil of turpentine, 

14 

— 10 

2 Tin, 1 Lead, . 



360 

181 

1 snow, 1 salt, 

0.00 

— 17.8 

1 Tin, 3 Lead, 



500 

260 

1 alcohol, 1 water, 

— 7 

— 21.6 

1 Tin. 1 Bismuth, 

• 


283 

140 

Cyangen,.... 

— 30 

— 34.4 

3T+2L+5B, . 



212 

100 

Mercury, 

— 40 

— 40 

1T + 1L+4B, 



200 

93 

Sulphuric ether, 

— 47 

- 43.9 

2T + 3L + 2B, . 



199 

92 

Sulphurous acid,. . 

—105 

— 76 

Slag of copper, 
Slag of tin, . 

• 


2462 

2402 

1350 

1318 

Nitrous oxide, . 

Nitric acid, . 

—150 

- 55 

— 101 









































384 


Expansion of Bodies by Heat. 


EXPANSION OF BODIES BY HEAT. 

All bodies in nature expand when heated, and contract when cooled. Solids 
vary but little by the difference in temperature; liquids vary more, but gases are 
extremely susceptible to the impression of heat and cold. 

There is a very singular fact connected 
with the expansion and contraction of 
substances at and near-the-temperature 
of fusion, which may be illustrated in the 
accompanying figure. 

Let A B represent-the absciss-axis of 
temperature, C D the ordinate axis of ex¬ 
pansion or contraction, and the origin 0 
the temperature of fusion, 0 A the tem¬ 
perature of the solid, and 0 B that of the 
liquid. 

Let a solid of volume and temperature 
at a be heated, it will expand until it 
reaches a maximum volume at b. after 
which it contracts toward the temperature of fusion 0. The temperature still 
increasing, the liquid will continue to contract until it reaches a minimum 
volume or maximum density at d, after which it will expand toward e. The lines 
a b 0 and Ode are parabolas, of which the absciss-axis A B passes through the 

focuses f. The formula for the parabola is y = x a , in which the exponent n depends 
upon the nature of the substance operated upon, and also whether it is linear or 
volume expansion, x representing the temperature and y the volume. 

Ice melts at 32° Falir., and the water reaches its maximum density at d = 39° fas 
now accepted, but d is nearer 40°). Ice reaches its maximum volume at b = 24°. 
Ice and water are of equal density at the temperatures 16°, 32° and 48°. ice 
generally floats in -water, because the difference in temperature is less than 32°. 
but if ice of less than 16° is put into water of more than 48°, it will sink. The 
sanv> phenomenon takes place with other substances; for instance, solid cast iron 
put into molten cast iron will float, but if the fluid cast iron is at a white heat, 
like that in a pneumatic furnace (Bessemer), the solid irou will sink. 

The following formulas are deduced from experiments which have not extended 
through the temperatures of fusioti, except that for water, page 392. 

Notation of Letters. 

L = linear expansion of solids and liquids, per degree Fahr., between any 
temperatures. 

I = linear expansion per degree between 32° and 212°, as contained in the 
accompanying table. 

D and d = absolute temperature in degrees Fahr. 

n = exponent of .expansion, which varies inversely with the rate of expansion 
of bodies. 


Exponent n 

1.04 

2.5 

2.6 

2.77 

14.1 

15.6 

for 

Water. 

Glass. 

Iron. 

Copper. 

Platinum. 

Mercury. 



Liuear expansion per degree from 32° to T° will be 
t l n - rT 

L — - i I). 

10580000 v 


Linear expansion per degree between any temperature is 



_ l_ 

105S00U0 



The linear expansion per degree multiplied by 2 will be the surface expansion. 
The linear expansion per degree multiplied by 3 will be the volume expansion. 


J 
































Dilatation or Expansion op Substances, 


385 


Dilatation or Expansion of Substances, 

Per Degree of Fahrenheit Scale. 


Tempera¬ 

tures. 

Solids. 


Linear, l. 

Surface, a. 

Volume, v 

32° to 212° 

) 


0.00000478 

0.00000956 

0.00001434 

212 

U 

392 

V Glass, 

• 

0.00000546 

0.00001093 

0.00001639 

392 

66 

572 

i 


0.00000660 

0.00001320 

0.00001980 

32 

a 

212 

| Wrought iron, . . 


0.00000656 

0.00001312 

0.00001968 

32 

u 

672 


0.00000895 

0.00001790 

0.00002686 

32 

u 

212 

Soft, good iron, . 

• 

0.00000680 

0.00001360 

0.00002040 

32 

u 

212 

Cast iron, .... 


0.00000618 

0.00001236 

0.00001854 

32 

u 

212 

Cast steel, 

• 

0.00000600 

0.00001200 

0.00001800 

32 

u 

212 

Hardened steel, . . 


0.00000689 

0.00001378 

0.00002067 

32 

a 

212 

| Copper, 


0.00000955 

0.00001910 

0.00002865 

32 

u 

572 

• 

0.00001092 

0.00002184 

0.00003276 

32 

u 

212 

Lead, . . . . 


0.00001580 

0.00003160 

0.00004740 

32 

<( 

212 

Gold, pure, . 

Gold, hammered, 

• 

0.00000815 

0.00001630 

0.00002445 

32 

<t 

212 


0.00000830 

0.00001660 

0.00002490 

33 

u 

212 

Silver, pure, 

Silver, hammered, . 

• 

0.00001060 

0.00002120 

0.00003180 

32 

u 

212 


0.00001116 

0.00002232 

0.00003348 

32 

a 

212 

Crass, common cast, . 
Crass, wire or sheet, 

• 

0.00001043 

0.00002086 

0.00003129 

32 

a 

212 


0.00001075 

0.00002150 

0.00003225 

32 

sc 

212 

| Platinum, pure, 


0.00000491 

0.000009S2 

0.00001473 

32 

66 

572 

• 

0.00000520 

0.00001040 

0.00001560 

32 

66 

212 

Palladium, 


0.00000555 

0.00001110 

0.00001665 

32 

66 

212 

Roman cement, . 
Platinum, hammered, 

• 

0.00000797 

0.00001594 

0.00002391 

32 

66 

212 


0.00000530 

0.00001060 

0.00001590 

32 

66 

212 

Zinc, pure or cast, 

Zinc, hammered, 

• 

0.00001633 

0.00003266 

0.00004899 

32 

6l 

212 


0.00001722 

0.00003444 

0.00005166 

32 

66 

212 

Tin, cast, . . . 

Tin, hammered, 

• 

0.00001207 

0.00002414 

0.00003621 

32 

66 

212 


0.00001500 

0.00003000 

0.00004500 

32 

66 

212 

Fire brick, . 

• 

0.00000275 

0.00000470 

0.00000705 

32 

66 

212 

Good red brick, 


0.00000305 

0.00000610 

0.00000915 

32 

66 

212 

Marble, 


0.00000613 

0.00001226 

0.00001839 

32 

66 

212 

Granite, . . . . 


0.00000438 

0.00000876 

0.00001314 

32 

66 

212 

Cismuth, . 


0.00000773 

0.00001546 

0.00002319 

32 

66 

212 

Antimony, 


0.00000602 

0.00001204 

0.00001806 

32 

66 

212 

Palladium, . 

• 

0.00000555 

0.00001110 

0.00001665 

32 

66 

212 

) 


0.00003333 

0.00006666 

0.00010000 

212 

66 

392 

y Mercury, . . 


0.00003416 

0.00006833 

0.00010250 

392 

66 

572 


0.00003500 

0.00007000 

0.00010500 

32 

66 

212 

) 


0.00008806 

0.00017612 

0.00026420 

212 

66 

392 

y Water, . . 

• 

0.00017066 

0.00034133 

0.00051020 

392 

66 

572 

f 


0.00018904 

0.00037808 

0.00056713 

32 

66 

212 

Salt, dissolved, 


0.00009250 

0.00018500 

0.00027780 

32 

66 

212 

Sulphuric acid, . 

• 

0.00011111 

0.00022222 

0.00033333 

32 

66 

212 

Turpentine and ether, 


0.00012966 

0.00025933 

0.00038900 

32 

66 

212 

Oil, common, 

Alcohol and Nitric Acid, 


0.00014814 

0.00029629 

0.00044444 

32 

66 

212 


0.01)015151 

0.00030302 

0.00055555 

32 

66 

212 

All permanent gases, 

• 

0.00069416 

0.00138832 

0.00208250 


Force of Temperature. 

It is the force of temperature which expands the bodies, and not the quantity of 
heat. See pages 379 and 392. Temperature is convertible into force. Let P 
denote the force of pressure in pounds per square inch, and 2’temperature Falir. 

Then, P = ( r+1Q5 - - - Y, and T= 202.8y /_ P—105.1. 

V 202.8 / 

This force, multiplied by tne space of expansion, is the work done by the heat. 


25 
















386 


Properties op Heat. 


Conducting Power of Different Substances for Heat and 

Electricity. 


Metals. 


Quartz sand, . . . 

35.56 

Liquids. 


Silver, fine, . 

100 

Limestone, .... 

19.8 

Water,. 

1.000 

Gold, “ . . 

98 

Lime,. 

24.00 

Mercury, .... 

2.80 

Gold. .991, . 

81 

Quartz crystals, . . 

80.0 

Proof spirit, . . . 

0.847 

Copper, ham’d, 

85 

Slate, . 

10.00 

Alcohol, pure, . . 

0.931 

Copper, cast, 

81 

Keen’s cement, . . 

1.901 

Nitric acid, . . . 

1.5 

Mercurv, . . 

6S 

Plaster and sand, 

1.870 

Sulphur, acid, . . 

1.7 

Aluminium, . 

66 

Plaster Paris, . . . 

2.026 

Sulphur, ether, . . 
Turpentine, . . . 

2.1 

Zinc, hammered 

64 

Roman cement, . . 

2.080 

3.1 

Zinc, cast ver- 


Asphalt,. 

4.52 

Gases. 


tical, . . . 

63 

Chalk,. 

5.853 

Air,. 

0.9S55 

Zinc, cast hori¬ 
zontal, . . 

Lead, cast, 
Cadmium, . . 

Wrought iron, 
Tin, .... 
Steel, . . . 
Platinum, . . 
Cast iron, . . 
Antimony, cast 
vertical, . . 
Antimony, cast 
horizontal, 
German silver, 
Bismuth, . . 

60 

20 

57 

43 

42 

40 

40 

36 

21 

19 

10 

6 

Woods. 


Radiating Power. 

100 

100 

98 

96 

95 

90 

88 

85 

80 

Fir, cross grain, . . 
Fir, with the fibre, . 
Pine, ...... 

Oak, with the fibre, 
Elm, “ “ . 

Ash, “ “ . 

Apple, “ “ . 

Ebony, “ “ 

Lampblack, . . . 

1.10 

3.10 

3.90 

3.30 

3.2 

3.1 

2.8 

2.2 
0.112 

Water,. 

Lampblack, . . . 
Paper, writing, . . 
Rosin, ..... 
Sealing-wax, . . . 
Glass, common, 

India ink, .... 

Ice,. 

Red lead, .... 

Cross: with fibre=l: 3, 

Birch,. 

Black oak, .... 
Chestnut, .... 

4.10 

3.2 

3.0 

Graphite, . . . 

Lead, tempered, 
Mercury, .... 
Lead, polished, . . 

75 

45 

20 

19 

Spanish mahogany, 

2.8 

Iron, polished, . . 

15 

Stone <£ Crystals. 


Walnut. 

3.3 

Tin and silver, . . 
Copper and gold, . 

12 

Marble, . . 

12.21 

Fur. 


12 

Glass, . . . 
Common brick, 

9.65 

ITare’s fur, .... 

0.0946 

Reflecting Powers. 


8.422 

Eider down, .... 

0.0668 

Brass,. 

100 

Fire-brick, 

6.05 

Beaver’s fur, . . . 

0.0675 

Silver,. 

90 

Fire-ela v, . . 

6.61 

Raw silk, .... 

0.0692 

Tinfoil,. 

85 

Porcelain, 

7.55 

Wool, sheep, . . . 

0.0778 

Tin, . 

80 

Wood-ashes, . 

0.8359 

Cotton,. 

0.0834 

Steel,. 

70 

Coal, anthracite 

19.25 

Lint,. 

0.0846 

Lead,. 

60 

Coal, bitum., . 

16.84 

Sewing-silk, . . . 

0.0955 

Glass,. 

10 

Coal, charred, 

0.738 

Flannel, .... 

0.395 

Glass, oiled or waxed, 

5 

Coke, . . . 

19.80 

Ilorse-hair, .... 

0.304 

Lampblack, . . . 

0 


Miscellaneous Temperatures. 


In the Bessemer furnace, 

Fahr. 

4000° 

1 alcohol, 1 water freezes, . 

Fahr. 

— 7° 

Puddling-furnace, 

3500 

Mean temp, of the poles, 

— 13 

Cupola,. 

3000 

Temp, outside atmosphere. 

— 58 

Heat of common fire, . 

1100 

Greatest natural cold, . 

— 56 

Red heat in daylight, 

1070 

Vinous fermentation, 

— 65 

Iron red in dark, . 

752 

Acetous fermentation begins, 

— 78 

Mean temp, of the earth, 

50 

Acetification ends, . . . 

— SS 

“ “ “ torrid zone, 

75 

Phosphorus burns, 

— 43 

“ “ ‘‘ temp. “ 

50 

Greatest artificial cold produced, 

—166 

“ “ “ polar region, 

20 

At 50°, Mixtures of- — 

Erod. 

Temp, of ignition, 

636 

Nitrate of ammonia, . 11 

cold. 

46 

Highest temp, of wind, . 

117 

Water,.1 j 

Temp, of the human blood, 

98 

Sulphate of soda, . . 81 

60 

A comfortable room, 

70 

Muriatic acid, . . . 5/ 

Mean temp, of ocean, . . 

62 

Dilate sulphuric acid, . 51 

23 



Snow, . . . . 4 J 




































Air and Heat. 


387 


ON AIR AND HEAT. 


Dry air expands or contracts uniformly 0.0020825 its volume per degree Fahr. in 
difference of temperature, or 0.0037485 per degree Centigrade under constant 
pressure. Assuming the expansion per degree Fahr. as unit, the primitive volume 
will be— 


1 

0.0020825 


= 480. 


F and v — volumes of dry air of temperatures t and 7 Fahr., and pressure^? and 
P above vacuum. The volumes and pressures in the following formulas maybe 
expressed in any units of measures. 


Volume and Temperature Variable under Constant Pressure. 

/T—t , , rn „ 430 (F—l) 

F== *VW +1 )’ and (T ~ i) = -7-> L 

Example 1. v = 18 cubic feet of air, of t = 36°, is to be heated to T= 84° under 
constant pressure. Required, the volume F? 

T ;r =18 -f-1^ =19.8 cubic feet, the answer. 

Volume and Pressure variable under Constant Temperature. 

( T — t) = 0, the pressures will he — 


p_ 

v_ 

and 

p V 

2. 

p 

V 


~p~ V ’ 

II 

^3 

V 

and 

X 

II 

3. 


, auu v - r , 

V P 


Example 2. V= 150 cubic feet of air, of pressure p — 1475 pounds to the square 
inch, is to be compressed to 50 cubic feet. The heat generated in the compression 
to radiate through the vessel until the temperature of the compressed air is equal 
to that in V. Required, the pressure P ? 

Formula^. P = p — = 14.75 X — = 44.25 lbs. 
v 50 


When tbe Temperature, Volume and Pressure 

are all variable, we have — 


?=«!(—+l\ and {T—t) 
* v \ 480 / 


480 SH- 4 - 

It must be distinctly understood in all these formulas that the volume v belongs 
to the pressure Pand temperature T, and the volume F to p and t. The prim¬ 
itive quantities are v, P, T. It may happen in the Formula 4 that r> V, <> T, 
aud p > P. 

Example 3. F= 1000 cubic inches of air of p = 14.75 and 2 = 59°, is to be 
reduced to r = 320 cubic inches, and the temperature increased to 2=369°. 
Required, tbe pressure P per square inch? 

^ . m -j-, 7., 1000 /369— 59 a „_ nll 

Formula 4. P=14./o——— (———-l)=7o.8 1bs. 

320 \ 480 J 

Example 4. v = 290, P= 88.5 to be increased to. V = 838, and p = 18.4. Re¬ 
quired, the difference of temperature ( T — t) ? 

Formula 4. ( T— t) = 480 (— ' 5 X 2 - ° — l) = 

v ; \ 18.4 X 838 } 


320 °. 
















888 


Air and Heat. 


On the Compression and Expansion 

of a definite weight of air enclosed in a vessel. 

In this treatment no heat must be lost or gained by radiation from the sides of 
the vessel in which the air is enclosed. Let D and d represent the degrees of 
absolute temperatures of volumes v and V of the air to be experimented upon. 

The absolute zero is 461° below Fahr. zero, and 274° Cent, below the freezing- 
point of water. D — 461 -f- T, d = 461 -j- t , and D — d= T — t, Fahr. scale. 


Volume and Temperature. 


V 


Compression D 


_ /£>\ 2 - 45 

~\d) ’ 

and 

-(f)“ 

n • ( d V" 

Compression v,= J 

2.45 | y 

d 

* V 

2*45 / yy 

Expansion d=D -y — 


5. 

7. 


Example 5. To what fraction must air of t — 65° be compressed, in order to fire 
tinder at a temperature of T— 550°, d = 461 -f 65 — 526°, D = 550 + 461 = 1011° ? 

V / 526 \ 2-45 

Formula 5. — = ( -) = 0.20, the answer. 

v Vioii/ 

Example 6. How much must air of T= 80° be expanded to reduce the temper¬ 
ature to £ = 32°, or freezing-point of water? 

V 1 541 \ 2-45 

Formula 5. —= (-) =1.3308 times, the answer. 

v 1493/ 

Example 7. v = 360 cubic inches of air of temperature T— 380°, or D =841°, 
is to be expanded until the temperature becomes t — 80° or d = 541°. Required, 
the volume F, corresponding to that temperature? 

( Q91 \2-45 

-) = 1025.9 cubic feet. 

541 / 


Example 8. V= 20 cubic feet of air of t = 32°, ord = 493, is to he compressed 
to u = 12 cubic feet. Required, the temperature T of compression? 

2.45 [20 

Formula 7. £> = 493 J— =60729°, or T= 146.29°. 

\ 12 


Pressure and Temperature. 


-=( 

P V 


d ) 


3>42 


and 


P_ 

F 


Compression 


/£) \ 3,42 


( d \ 342 

Expansion p = F ^ — i 


3.42 


P 

Compression D = d ^—, Expansion d = F 


I)) 

3.42 


P 




10 . 


Example 9. A volume of air of pressure p — 15 pounds to the square inch, and 
of temperature < = 62°, is to be compressed until the temperature becomes 
T~ 120°. Required, the pressure P per square inch at T = 120° ? 

d = 461+62 = 523, and £> = 461 + 120 = 581. 

/ 581 \ 3,42 

= 21.49 lbs. pr. sq. inch. 


Formula 9. 


P= 15 (- 




V 523 7 




389 


Air and Heat. 

■ “ -— c - 

Example 10. A volume of air of pressure P=45 pounds to the square inch, 
and of temperature l 1 ==. 250°, or D = 711°, is to be expanded to a pressure of 
P — 25 pounds. Required, the temperature t of the expanded air ? 


3.42/ OK 

Formula 10. cZ = 711 /—= 598.72°, and 

\ 45 5 

t — 598.72 — 461 = 137.72°, the temperature required. 


Pressure and Volume. 


2.45/ Y 3.42,— 

\T~X~p 

i.4rp 

Expansion V~v —, 

\ P 

Mfr 


and 


2-45 l~V_ 
V 


Compression v = V^j 


Compression 


, Expansion 


rfi. 

. 11. 



• 

> 

. 12. 

, \ 1-4 

. 13. 


Example 11. A volume v = 50 cubic inches, and of pressure P= 80 pounds per 
square inch, is to be expanded until the pressure becomes p — 15 pounds. Required, 
the expanded volume V? 

lA i~S0 

Formula 12. V — 50' /— =165 cubic inches. 

\ 15 

Example 12. What will be the pressure of a volume of air expanded 1.3308 
times ? 

1.4 


Formula 13, 


r \ 1.3308/ 


0.5324 of the primitive pressure. 


"Volume and Weight of Pry Air 

At different Temperatures, under a constant Atmospheric Pressure of 29.92 inches 
in the Barometer, the Volume at 32° Fahr. being the unit. 


Temp. 

Fahr. 

Volume. 

Wt. per 
Cub. ft. 
Pounds. 

0° 

.935 

.0864 

12 

.960 

.0842 

22 

.980 

.0824 

32 

1.000 

.0807 

42 

1.020 

.0791 

52 

1.041 

.0776 

62 

1.061 

.0761 

72 

1.082 

.0747 

82 

1.102 

.0733 

92 

1.122 

.0720 

102 

1.143 

.0707 

112 

1.163 

.0694 

122 

1.184 

.0682 

132 

1.204 

.0671 

142 

1.224 

.06 "9 

152 

1.245 

.0649 


Temp. 

Fahr. 

Volume. 

Wt. per 
Cub. ft. 
Pounds. 

162° 

1.265 

.0368 

172 

1.425 

.0628 

182 

1.306 

.0618 

192 

1.326 

.0609 

202 

1.347 

.0600 

212 

1.367 

.0591 

230 

1.404 

.0575 

250 

1.444 

.0559 

275 

1.495 

.0540 

300 

1.546 

.0522 

325 

1.597 

.0506 

350 

1.648 

.0490 

375 

1.689 

.0477 

400 

1.750 

.0461 

450 

1.852 

.0436 

500 

1.954 

.0413 


Temp. 

Fahr. 

Volume. 

Wt. per. 
Cub. ft. 
Pounds. 

550° 

2.056 

.0384 

600 

2.150 

.0376 

650 

2.260 

.0357 

700 

2.362 

.0338 

800 

2.566 

.0315 

900 

2.770 

.0292 

1000 

2.974 

.0268 

1100 

3.177 

.0254 

1200 

3.381 

.0239 

1500 

3.993 

.0202 

1800 

4.605 

.0175 

2000 

5.012 

.0161 

2200 

5.420 

.0149 

2500 

6.032 

.0133 

2800 

6.644 

.0121 

3000 

7.051 

.0114 


For Weight and Volume of air at Low Temperature, see Ilygrometry, page 357. 


























990 


SrEciFig Heat. 


SPECIFIC HEAT OR CALORIC. 


Different bodies require different quantities of heat for equal difference in tem¬ 
perature. This difference is called specific heat. The specific heat of bodies varies 
nearly inverse as the specific gravity. The specific heat in all bodies increases 
slightly with the elevation of temperature. One pound of water elevated from 
32° to 212° requires 180.9 units of heat instead of 180. The specific heat increases 
nearly in the same ratio for all solid and liquid bodies. The specific heat of water 
from *32° to T° will be— 


S = l + 


(T - 32) 1 - 67 
1167713 


1 . 


The specific heat of water between any temperatures T and t will be— 

„ - , (T - 32) 1 * 67 — (t - 32) 1 * 67 

O - i- ” I • 

1167713 


2 . 


The following table gives the specific heat of different substances between the 
temperature 32° and 212°, compared with water as unit. When the specific heat 
of a body is required between high temperatures, it is necessary to calculate first 
the specific heat of water between such temperatures, which multiplied by the 
number in the table will give the required specific heat of the body. 


Specific Heat or Caloric of Substances. 


Water, .... 

1.000 

Lead, .... 

0.030 

Sweet oil, . . . 
Oil of turpentine, 

0.310 

Ice of water, 

0.513 

Steel,. 

0.118 

0.472 

Cast iron, .' . . 
Wrought iron, . 

0.140 

Diamond, . . . 

0.147 

Gases of constant 


0.110 

Arsenic, .... 

0.081 

volume and under 


Cobalt, .... 

0.150 

Iodine, .... 

0.054 

atmospheric pres- 


Nickel, .... 

0.103 

Sulphur, .... 

0.200 

sure. 


Copper. 

Zinc,. 

0.094 

Lime, burned, . 
Glass-crystals, . . 

0.217 

Atmospheric air, . 

0.250 

0.093 

0.193 

Oxygen, . . . 

0.230 

Tin,. 

0.047 

Glass, common, 

0.177 

Hydrogen, . . . 

3.30 

Antimony, . . 

0.051 

Woods, average, . 

0.500 

Nitrogen, . . . 
Carbonic acid, . . 

0.275 

Bismuth,.... 

0.030 

Brick, common, 

0.200 

0.221 

Tellurium, . . 

0.091 

Firebrick, . . . 

0.220 

Carbonic oxide, . 

0.288 

Gold,. 

0.029 

Coal,. 

0.261 

Olefiant gas, 

0.421 

Silver, .... 

0.057 

Beeswax,.... 

0.450 

Nitro-oxide, . . 

0.237 

Platinum, . . . 

0.034 

Alcohol, s. g. 0.81, 
Sulphuric acid, 

0.700 

Gas of oils, . . 

0.421 

Brass, .... 
Mercury, .... 

0.094 

0.335 

Sulp. hydrogen, . 

0.242 

0.030 

Nitric acid, . . 

0.661 

Steam of atm. pr., 

0.475 


Let two different substances of known weight or volume and temperature be 
mixed together; the temperature of the mixture will dissolve the relative quantity 
of caloric in the ingredients. 


Mixture of the same Substances. 

W= weight or volume of a substance of temperature T. 

.w = weight or volume of a similar substance of temperature t. 
t = temperature of the mixture W + w. We shall have— 


t'[ W-\-w)= WT-\- Wt , 3. 


_ w{t' — t ) 

T - t' ’ 


t' =■ 


T 


WTwt 
W-\-w ’ 
w{t' — t) , 

W~ 1 


t'. 


5. 

6 . 




























Specific Heat. 


391 


Example 1. Let W— 4.62 cubic feet of water at T— 150° be mixed with w = 6.43 
cubic feet at t = 46. Required, the temperature of the mixture V — ? 

„ 4.62 X 150° + 5.43 X 46° 

' =-462 T'5.43-= 97 ' 8 > the anSWer ' 


Example 2. How much water of T— 107° must be mixed with w ■■ 
of t = 58°, the mixture of the water to be 75° ? 


27.3 gallon 


y= grj»(75-S8 ) ==m 


107 — 75 

Mixture of Different Substances. 

W and w expressed by weights only. S and s = specific caloric as given in the 
accompanying table. We shall have— 


W S( T — t') =ws{V — t), 7. 
//(+ —wst 




WS 


8 . 


t' =■ 


W= 


W ST-\-wst 

TF/S'-f- ws 
ws{t r — t) 

s{T—ty 


10 . 


Example^. To what temperature must W— 20 pounds of cast iron be heated 
to raise w = 131 pounds of water of t — 54° to a temperature t' — 64° ? T— ? 
From the table we have s = 1, and S= 0.14. 


T= 


64(20 X 0-14 + 131) — 131 X 1 X 54 
20 X 0.14 


= 532°, 


the required temperature, supposing no vapor escapes from the water. 

If any chemical action takes place in the mixture, these formulas will not an¬ 
swer, because part of the sensible caloric may become latent , or latent caloric may 
be set free. 

Example 4. The temperature of 5 pounds of copper is to be elevated from 60° 
to 80°. How many calorics will be required ? 

See table for copper 0.094 (80 — 60) = 1.88 calorics, the answer. 

Specific Heat of Gases. 

When heat is applied to a constant volume of gas enclosed in a vessel, the 
specific heat of that gas increases as the square root of the pressure generated by 
the heat. 

s =™m. a. 

Vp 

When the volume, pressure and temperature are all variable, the specific heat 
of air will be— 

»■“« ,, 




l 493 

For any permanent gas of s — specific heat under atmospheric pressure, the 
specific heat under any other pressure and volume will be— 

S = - 3 8348 13. 


y V \ 493 I 




















892 


Dynamics and Units of Heat and Work. 


The weight of 320 cubic reet of air at 59° is (see table) 0.076 X 320 = 24.32 
pounds. The calorics consumed in the operation will be— 

Formula 13. h = 1.08 X 24.32 (369 — 59) = 1050.625 units. 

The specific heat of any other gas, under different volumes, pressures and tem¬ 
peratures, is equal to that of air multiplied by the specific heat in the table and 
divided by 0.25. 

The number of calorics h required to elevate the temperature of W pounds of 
gas from t° to T°, will be— 

h = SW(T — t) . 13. 

When the pressure is constant, and the volume is increased by heat, and S= 
specific heat of the primitive volume of air, then the calorics will be— 


h = SW{T—t) 


5.36 


(t-> 


14. 


Formula 14. 


in which the last term expresses the calorics expended in expanding the volume 
under the pressure P. 

Example 6. One cubic foot of air of t= 32° is enclosed in a cylinder of one 
square foot area of piston, and under atmospheric pressure P= 14.76 pounds to the 
square inch. Let heat be applied to the air until 7 7 =511°, when the volume will 
be about double, the piston being well balanced to move with the constant pres¬ 
sure. Required, the number of calorics imparted to the air, and the heat ex¬ 
pended in moving the piston with the pressure 14.75 pounds. 

h = 0.25 X 0.0807 X (511—32) + — 1^ = 9.65 + 2.75 = 

12.4 calorics, of which 2.75 was expended in moving the piston one foot. 

DYNAMICS AND UNITS OF HEAT AND WORK. 

The ordinary English unit of heat is that required to elevate the temperature 
of one pound of distilled water one degree Fahr. from 39° to 40° (Fahr. lb.), and 
called one caloric. 

The German unit of heat is that required to elevate the temperature of one 
pound (German pfund) of water one degree Centigrade, from 4° to 5° (Cent, lb.) 
or (Cent, pfd.) 

The French unit of heat (called calorie) is that required to elevate the tempera¬ 
ture of one kilogramme (2.2047 lbs.) of water one degree Centigrade from 4° to 5° 
(Cent. kilo.). 

A combination of French and English units of heat is sometimes expressed by 
Fahr. scale and French weight (Fahr. kilo.). 

Heat is dynamic work, or the product of the three simple elements, force , 
velocity and time, in which the temperature of the heat represents force , and the 
cubic contents of the units of heat represent the product of time and velocity , 
which is space. 

The English unit of dynamic work is one pound raised one foot, called footpound 
(Ft. lb.). One caloric = 772 ft. Ibs. 

The French unit of dynamic work is one kilogramme raised one metre, called 
kilomet. 

One horse-power will consume or generate 2564 calorics per hour. 

Comparison of Different Units of Heat and Worlc. 


English Calorics. 

French Calorie. 

Prussian. 

Dynamic Work. 

Fahr. 2>. 

Cent. B>. 

Fahr. kilo. 

Cent. kilo. 

Cent. pfd. 

Ft. a. 

Kilomet. 

1 

0.5555 

0.4536 

0.2520 

0.5769 

772 

106.51 

1.8 

1 

0.8165 

0.4536 

1.0385 

1389.6 

191.71 

2.2047 

1.2248 

1 

0.555 

1.2719 

1702 

348.066 

3.968 

2.2047 

1.8 

1 

2.2894 

3063.6 

626.52 

1.733 

0.9630 

0.7862 

0.4368 

1 

1368.2 

273.66 

.0912953 

.0007196 

.0005876 

.0003264 

.0007473 

1 

0.13825 

.0093896 

.0005205 

.0004250 

.0002361 

.0005405 

7.233 

1 












JXTSS AND Gt2fJVWn>KR. 


$» 


IVnonuam e, \Y eiglit aud Dimensions of Heavy Ordnance. 


tasotimw. 

Riant 

i 

Rore. 

l.ensrth 

f 

Gun. 

W eight in Pounds of 

Gum Proj'tile Powd. 

•J 

Bore. 


VV.C » 

ft. ia. 

.:a,a. 

IViu.is. 

r.'-awAs 

Ft. jvx sec 


American, rife, . . 

4* 

7 r 

8. o80 




Rifle. 

O M 

* * 

0 

r r* 

7.070 


* • 

♦ ♦ * 

Rifle. 

Rug.. -wrought iron. 

8 

o i<r 

u.chy 

180 

SO 

1854 

Smooth. 

.Vrr-.-.v.- css: r.r. 

$ 

W 

0.084 

... 

• * 

» * » 


English. 

10 

ir 

4mS50 

400 

00 

1508 

a 

American, * 

11 

IV 

10.811 



♦ ♦ ♦ 

Ot 

Russian. ** 

U 

11 O'* 

88.800 

400 

85.8 

1885 

u 

English. * 

15 

15* 

,\\S00 

000 

87 

1180 

o 

American, 

18 

IS* 8 V 

10.811 

a * ♦ 



it 

n a 

18 

lo' fT 

45,101 

... 

* % 

% * v 

it 

a* h 

5;' 

S’* 

118.VW 

90S 

150 

1181 

ot 

*■ mortar, . 

IS 

5 R*** 

17.108 

« ♦ % 


... 

it 

Nns. brass. M scow. 

30 

58’ 

80.vV0 

8000 

% « 

, . . 

Ot 


ESVfl of Unu}Hmder, 

The :‘v",v.v.\' work of different k-:nls of dor, utilised in heavy ordnance, 

var.-s between l.WCvY and 5vVaW loot'ponnds ivr ponnd of powder. Let .V de¬ 
note the d,y v.srotv w ork >u a charge of powder. 0 weight in pounds of the hall 
or projectile, V velocity of the projectile iu toot per sovud, thru 

, V _ UK , „ 111 ’ 

r= ^¥- 1,=_ n’ Wld a= “5T" 

The length of the gnu ftvr these formulas should he at Lwt 15 times the here. 

Korre of CJunpowtlw, 

The force of gunpowder depends much upon Its quickness of burning and resist' 
arce to its expansion. Gunpowder eneKwed in a strong vessel, and burned in its 
r: o.t-.ve volume, msv reach a pressure of 100 tons to the square inch; hut when 
the gas of -powder is subjected to an excessive pressure, it gets cracked, as it were, 
ami o\ses the property of expansion due to a permanent gas less strained, t'his is 
a very important fact in the use of heavy ordnance, where the gun max he double 
strained with a loss of effect in the projectile. 

Quick powder may strain a gun over 80 tons to the square inch, w hilst slower 
powder will strain it only to tons, and give a greater velocity tv' the projectile. 

It appears that the charge ought tv' he so arranged in a gnu that a slow powder 
s irst ignited. and then a quicker and quicker until the quickest at last, by winch 
the gun need not he strained tv' more than 18 tons to the square inch, with full 
benefit of the expansion pt\»porty of the gas, greater velocity of the projectile and 
less risk in hutsting the gun. The work done by the gas v't powder in a gun should 
be treated under the same laws as that of steam in a steam cylinder, 

l'his special subject is too extensivo for proper treatment in this Pocket book. 
Composition of (InnpotvtU'r, 

The composition of gunpowder varies in all proportions hetweon the limits of 
70 am; 7S parts of saltpetre ^nitratoof potash, IvO, Nv\„\ 18 and 18 putts oi chatVvol, 
0 ami 50 parts of sulphur in 100 parts of the powder, 

Chinese powder, d'J saltpouv, *J> oUtmval, lo sulphur. 

The ditferont proportions depend much upon the purpvvso for w hich the powder 
is used, and also upon the ivleas and experience v't the inunuthctuwru ami users of 
the powder. The quickest pow der requires the lOgluxt peopovWous v't saltpetre, 
Slir of (luupowtter^rniiu 

V\'MlWO\vVN 

77,18,10 
Ttkl8.ll 
TM3,15 
74 , 11,15 


I'SR or POWl'KR. 

Sur or StRYR, 

Si s r w Incurs. 

Sporting,. 

No, l to 5, 

0,08 to 0,0fl 

Mortar, 

No. 5 to 8, 

O.tH'v to iU 

Cannon, . 

N\v* 4 to t\ 

0,58 to 0.88 

Mammoth, . 

Mo, 8 to T. 

0,0 to 0,0 


Kino gunpowder is also moulded into lumps to tit (lie cha 
The Russians mould tine pvnvdev intv' hexagon blocks fv'r 


heavy ordnance. 




































894 


Properties op Water and Steam. 


PROPERTIES OF WATER AND STEAM, 

In Relation to Heat. 

The following six pages of tables for water and steam have been calculated by 
the author whilst stationed in the Bureau of Steam Engineering of the United 
States Navy Department, under the direction of Chief Engineer Isherwood. The 
tables have been improved for this Pocket-book. 

Properties of Water. 

Column h' contains the calorics required to raise each cubic foot of distilled 
water from 32° to temperature T, under the pressure P. 

Column h contains the calorics required to raise each pound of water from 32° 
to T°. This column is calculated from the formula deduced from Reguault’s 
experiments, namely: 

A -r°- 8 2° + < r °-32)" , mV _ T _ ^T-wsr-w-^ L 

p / p / 

in which the last term is a parabola of exponent n — 2.67, and parameter p' 1167713. 
log. p' = 6.0673350. h' = calorics required per pound of water of temperature t', 
and raised to T°. 

Column c contains the fractional cubic feet per pound of water of temperature T. 

Column w contains the weight in pounds per cubic foot of water of temperature 
T. Water of the maximum density at 39° weighs 62.388055 pounds per cubic foot. 

Column v contains the volume of water of temperature T, that at 39° being 
unit. This column is calculated from the Formula 2, deduced from Kopp’s experi¬ 
ments. 

_ {T — 39) 2 _ 

2000000 [0,23 + 0.0007 (T —39)] * 


v — l-\ 


Column t contains the temperature of the steam and water, Celsius’ scale. 

Columns i and p give the steam-pressure indicated on the safety-valve or 
mercury-gauge. 

-f means pressure above the atmosphere. 

— means vacuum under the atmosphere. 

Properties of Steam. 

Column P contains the total steam-pressure in pounds per square inch, in¬ 
cluding the pressure of the atmosphere. 

Column I is the same pressure in inches of mercury, The specific gravity of 
mercury at 32° Fahr. is 13.5959, compared with water of maximum density at 39°. 
One cubic inch of mercury weighs 0.49086 pounds, of which a column of 29.9218 
inches is a mean balance of the atmosphere, or 14.68757 lbs. per sq. in. 

Column T contains the temperature of the steam on Fahr.’s scale, deduced from 
Regnault’s experiments. 

Column V contains the volume of steam of the corresponding temperature T , 
compared with that of water of maximum density at 39° Fahr. This column is 
calculated from the formula of Fairbairn and Tate, namely: 


V— 25.62 + 


49513 
/+ 0.72 ‘ 


3. 


Column IF contains the weight per cubic foot in fractions of a pound; and 
Column G the cubic feet per pound of saturated steam under the pressure P 
and temperature T. 

Column H contains the calorics per pound of steam from 32° to temperature T 
and pressure P, calculated from the formula— 


H = 1081.91 -f- 0.305 T. . ... 4. 


Column H r contains the calorics per cubic foot of steam from 32° to tempera¬ 
ture T. 


J 








Properties of Water and Steam. 


395 


The columns H and II' give the calorics required to heat the water from 32° to 
the boiling-point and evaporate the same to steam under the pressure P and of 
temperature T. 

Column L contains the latent units of heat per pound in steam of temperature 
T and pressure P. The latent heat expresses the work done in the evaporation, 
or the difference between the calorics per pound in the steam and in the water of 
the same temperature. 

Column L' contains the latent heat per cubic foot of steam. 

Latent heat L = H — h, the calorics required to evaporate each pound of water 
from the boiling-point into steam. 

The maximum work K, which can be realized per caloric in steam without 
expansion, is— 


K= 


144 P{V — 1) 
IV V 


. 5. 


50.14 footpounds, 


Example 1. Required, the maximum work K—l that can be realized per caloric 
in steam of P — 50 lbs. per sq. in. ? V = 508.29 and H' = 143.3. 

R _ 144 X 50 (508.29 — 1) 

~~ 143.3 X 508.29 

or, 50.14 : 772 = 0.0649 of the natural effect. 

The maximum work which can be realized per caloric in steam with expansion 
will be— 

144P(F —1)(2.3 log.- + l) 

K = ---, . . 6 . 

IVV ’ 

in which S — stroke of piston, and l = part of the stroke with full steam. 

NIC 

The natural effect of a steam-engine in horse-power is = - 

r550 


of which from 50 to 75 per cent, is realized in ordinary practice. JY — numler of 
calorics passed through the engine in the time t in seconds. 

Example 2. Let the steam in Example 1 be expanded S: 1 = 3 times. We 
have log. 3 = 0.47712, and 2.09737 X 50.14 = 105.16 footpounds per caloric. Sup¬ 
pose each stroke of the piston to use 4 cubic feet of steam expanded 3 times, and 
making 90 strokes per minute. 


Then 


90 X 4 X 143.3 


and the power will be 


60 

439.8 X 105-16 
1 X 550 


439.8 calorics per second, 
84 horses. 


This is the effect of steam when raised from water of 32°, but when the feed- 
water is of higher temperature, calculate the calorics from the Formula 1, h', and 
add the latent heat per pound of the steam; the sum will be the calorics required 
in generating the steam. 

Tlie late Professor Alexander, of Baltimore, gave a very simple 
and clear formula for temperature and pressure of steam, which may be as 
reliable as the experiments of Regnault, from which it differs very little, namely : 

J=(— + —)\ and T= 180 105.13. 

V180 1695/ 


The inches of mercury I X 0.48875 = pressure in pounds per square inch. 
T — temperature of the steam, Fahr. 

The pressure in atmospheres A will be— 


A = (—x 5(iI —V and T = 317.13 $ A — 105.13. 

\ 317.13 169.5 / 












Properties op Water from Freezing to Boiling Point. 


3 "i 5 


Temp. 

Volume 

Units of heat. 

Pounds 

Cubic ft. 

Temp. 

Fahr. 

1 at 39° 

pr. lb. 

pr. cub. ft. 

pr. cub. ft. 

pr. lb. 

Celsius. 

rjlO 

V 

h 

h' 

VJ 

c 

t 

32 

1-000109 

0-000000000 

0-00000 

62.387 

0-01603046 

0-000 

33 

1-000077 

1-000000867 

62-383 

62-383 

0-01602994 

0-555 

34 

1-000055 

2-000000545 

124-77 

62-384 

0-01602956 

1-111 

35 

1-000035 

3-00001609 

187-16 

62-385871 

0-01602927 

1-666 

36 

1-000020 

4-00003468 

249-55 

62-386791 

0-01602904 

2-222 

37 

1-000009 

5-00006294 

311-99 

62-387493 

0-01602886 

2-777 

38 

1-000002 

6-00010241 

374-33 

62-387930 

0-01602874 

3-333 

39 

1-000000 

7-00015455 

436-72 

62-388055 

0-01602871 

3-888 

40 

1-000002 

8-00022076 

499-12 

62-387930 

0-01602874 

4-444 

41 

1-000009 

9-00030234 

561-51 

62-387493 

0-01602886 

5-000 

42 

1-000019 

10-00040056 

623-89 

62-386869 

0-01602902 

5-555 

43 

1-000034 

11-00051663 

686-28 

62-385933 

0-01602926 

6-111 

44 

1-000053 

12-00065175 

74S-66 

62-084748 

0-01602956 

6-666 

45 

1-000077 

13-00080704 

811-03 

62-383251 

0-01602994 

7-222 

46 

1-000104 

14-00098362 

873-40 

62-381567 

0-01603038 

7-777 

47 

1-000136 

15-001326 

935-70 

62-379571 

0-01603088 

8-333 

48 

1-000171 

16-0014050 

997-77 

62-377388 

001603146 

8-888 

49 

1-000211 

17-0016518 

1060-0 

62-374893 

0-01603210 

9-444 

50 

1-000254 

18-0019242 

1122-8 

62-372212 

0-01603278 

10-000 

51 

1-000302 

19-0022230 

1185-1 

62-369219 

0-01603355 

10-555 

52 

1-000353 

20-0025493 

1248-0 

62-366039 

0-01603437 

11-111 

53 

1-000408 

21-0029241 

1310-1 

62-362611 

0-01603525 

11-666 

54 

1-000468 

22-0032880 

1372-3 

62-358871 

0-01603621 

12-222 

55 

1-000531 

23-0037024 

1434-3 

62-354944 

0-01603723 

12-777 

56 

1-000597 

24-0041479 

1496-4 

62-350831 

0-01603828 

13-333 

57 

1-000668 

25-0046256 

1558-6 

62-346407 

0-01603942 

13-888 

58 

1-000740 

26-0051362 

1620-9 

62-341921 

0-01604057 

14-444 

59 

1-000819 

27-0056808 

1683-2 

62-337000 

0-01604184 

15-000 

60 

1-000901 

28-0062600 

1745-5 

62-331893 

0-01604316 

15-555 

61 

1-000986 

29-0068749 

1807-8 

62-326620 

0-01604451 

16-111 

62 

1-001075 

30-0075263 

1870-1 

62-321059 

0-01604594 

16-666 

63 

1-001167 

31-0082149 

1932-4 

62-315333 

0-01604741 

17-222 

64 

1-001262 

32-00S9416 

1994 4 

62-309420 

0-01604894 

17-777 

65 

1-001362 

33-0097073 

2056-6 

62-303198 

0-01605054 

18-333 

66 

1-001464 

34-010513 

2118-7 

62-296852 

0-01605218 

18-888 

67 

1-001570 

35-011359 

2180-8 

62-290259 

0-01605388 

19-444 

68 

1-001680 

36-012246 

2242-9 

62-283418 

0-01605564 

20-000 

69 

1-001793 

37-013175 

2305-0 

62-276293 

0-01605748 

20-555 

70 

1-001909 

38-014148 

2367-1 

62-269183 

0-01605921 

21-111 

71 

1-002028 

39-015164 

2429-2 

62-261788 

0-01606122 

21-666 

72 

1-002151 

40-016224 

2491-2 

62-254146 

0-01606318 

22-222 

73 

■ 1-002277 

41-017330 

2553-2 

62-246320 

0-01606521 

22-777 

74 

1-002406 

42-018482 

2615-2 

62-238309 

0-01606728 

23-333 

75 

1-002539 

43-019680 

2677-1 

62-230052 

0-01606941 

23-888 

76 

1-002675 

44-020926 

2739-2 

62-221612 

0-01607158 

24-444 

77 

1-002814 

45-022220 

2801-0 

62-212987 

0-01607382 

25-000 

78 

1-002956 

46-023563 

2862-8 

62-204179 

0-01607610 

25-555 

79 

1-003101 

47-024956 

2924-6 

62-195187 

0-01607841 

26-111 

80 

1-003249 

48-026398 

2985-4 

62-186012 

0-01608078 

26-666 

81 

1-003400 

49-027893 

3048-2 

62-176654 

0-01608321 

27-222 

82 

1-003554 

50-029438 

3111-0 

62-167113 

0-01608567 

27-777 

83 

1-003711 

51-031039 

3172-8 

62-157388 

0-01608820 

28-333 

84 

1-003872 

52-032688 

3234-4 

62-147420 

0-01609077 

28*888 

85 

1-004035 

53-034394 

3296.2 

62-137330 

0-01609338 

29-444 

86 

1-004199 

54-036154 

3358-2 

62-127182 

0-01609601 

30-000 

87 

1-004370 

55-037969 

3418-7 

62-116605 

0-01609875 

30-555 

88 

1*004542 

56-039841 

3480-4 

62-105969 

0-01610151 

31-111 

89 

1-004717 

57-041769 

3542-1 

62-095152 

0-01610432 

31-666 

90 

1-004894 

58-043754 

3603-8 

62-084214 

0-01610715 

32-222 

















Properties of Water from Freezing to Boiling Point. 


?97 


Temp. 

Volume 

Units of heat. 

Pounds 

Cubic ft. 

Temp. 

Fahr. 

1 at 39° 

pr. lb. 

pr. cub. ft. 

pr. cub. ft. 

pr. lb. 

Celsius. 

jro 

V 

h 

h' 

w 

c 

t 

91 

1-005094 

59-045797 

3665-0 

62-071860 

0-01611036 

32-777 

92 

1-005258 

60-047899 

3726-6 

62-061734 

0-01611298 

33-333 

93 

1-005444 

61-050061 

3788-2 

62-050252 

001611597 

33-888 

94 

1-005633 

62-052282 

3849-8 

62-038591 

6-01611900 

34-444 

95 

1-005825 

63-054564 

3911-2 

62-026749 

0-01612208 

35-000 

96 

1-006019 

64-056907 

3972-6 

62-014787 

0-01612519 

35-555 

97 

1-006216 

65-059312 

4033-9 

62-002646 

0-01612834 

36-111 

98 

1-006415 

66-061780 

4035-2 

61-990386 

0-01613153 

36-666 

99 

1-006618 

67-064311 

4156-5 

61-977885 

*0-01613478 

37-222 

100 

1-006822 

68-066906 

4217-7 

61-965322 

0-01613806 

37-777 

101 

1-007030 

69-069565 

4278-9 

61-952528 

0-01614140 

38-333 

102 

1-007240 

70*072290 

4340-1 

61-939612 

0-01614475 

3S-888 

103 

1-007553 

71-075080 

4401-3 

61-920370 

0-01614977 

39-444 

104 

1-007668 

72-077937 

4462-5 

61-913303 

0-01615161 

40-000 

105 

1-007905 

73-080861 

4523-0 

61-898745 

0-01615541 

40-555 

106 

1-008106 

74-083852 

4585-0 

61-886403 

0-01615863 

41-111 

107 

1-008328 

75-086912 

4645-9 

61-872778 

0-01616220 

41-666 

108 

1-00S554 

76-090044 

4706-8 

61-S58913 

0-01616581 

42-222 

109 

1-008781 

77-093239 

4767-7 

61-844994 

0-01616946 

42-777 

110 

1-009032 

78-096509 

4828-6 

61-829609 

0-01617348 

43-333 

111 

1-009244 

79-099846 

4889-5 

61-816622 

0-01617677 

43-888 

112 

1-009479 

80-103255 

4950-4 

61-802231 

0-01618064 

44-444 

113 

1-009718 

81-10674 

5011-3 

61-787602 

0-01618447 

45-000 

114 

1-009956 

82-11029 

5072-2 

61-773042 

0-01618829 

45-555 

115 

1010197 

83-11392 

5133-0 

61-758305 

0-01619216 

46-111 

116 

1-010442 

84-11762 

5193-7 

61-743331 

0-0161960S 

46*666 

117 

1-010688 

85-12140 

5254-3 

61-728302 

0-01620003 

47-222 

118 

1-010938 

86-12525 

5314-9 

61-713037 

0-01620403 

47-777 

119 

1-011189 

87-12918 

5375-5 

61-697719 

0-01620806 

48-333 

120 

1-011442 

88-13318 

5436-1 

61-682286 

0-01621211 

48-888 

121 

1-011698 

89-13726 

5496-6 

61-666678 

0-01621621 

49-444 

122 

1-011956 

90-14141 

5557-1 

61-650956 

0-01622034 

50-000 

123 

1-012216 

91-14565 

5617-6 

61-635123 

0-01622451 

50-555 

124 

1-012478 

92-14996 

567S-1 

61-619170 

0-01622871 

51-111 

125 

1-012743 

93-16435 

5738-6 

61-603047 

0-01623296 

51"666 

126 

1-013010 

94-15882 

5798-9 

61-586S10 

0-01623724 

52-222 

127 

1-013278 

95-16338 

5859-2 

61-580516 

001624153 

52-777 

128 

1-013550 

96-16801 

5919-5 

61-553998 

0-01624590 

53-333 

129 

1-013823 

97-17272 

5979-7 

61-537423 

0-01625027 

53-888 

130 

1-014098 

98-17752 

6040-0 

61-520735 

0-01625468 

54-444 

131 

1-014358 

99-18239 

6100-2 

61-504966 

0-01625884 

55-000 

135 

1-015505 

103-20274 

6340-3 

61-435497 

0-01627724 

57-222 

140 

1-016962 

108-23009 

6639-6 

61-347282 

0-01630064 

60-000 

145 

1-018468 

113-25965 

6937-9 

61-256765 

0-01632473 

62-777 

150 

1-020021 

118-29147 

7215-1 

61-163500 

0-01634961 

65-555 

155 

1-021619 

123-32562 

7531-2 

61-067829 

0-01637523 

6S-333 

160 

1-023262 

128-36217 

7826-2 

60-969776 

0-01640156 

71-111 

165 

1-024947 

133-40119 

8098-1 

60-869542 

0-01642857 

73-888 

170 

1-026672 

138-44273 

8412-8 

60-767270 

0-01645623 

76-666 

175 

1-028438 

143-48687 

8704-2 

60-662047 

0-01648477 

79-444 

180 

1-030242 

148-53666 

8994-9 

60-556699 

0-01651345 

82-222 

185 

1-032083 

153-58316 

9281-9 

60-448679 

0-01654296 

85-000 

190 

1-033960 

158-63545 

9571-6 

60-338944 

001657305 

87-777 

195 

1-035873 

163-69057 

9858-5 

60-227513 

0-01660370 

90-555 

200 

1-037819 

168-74858 

10318 

60-114581 

0-01663489 

93-333 

205 

1-039798 

173-80956 

10428 

60-000168 

0-01666662 

96-111 

210 

1-041809 

178-87355 

10712 

59-884350 

0-01679885 

98-888 

212 

1-042622 

180-90000 

18824 

59-837654 

0-01681160 

100-000 


l 

















898 


Properties of Water. 





Water. 



Indie. 

press. 

Temp. 

Units of heat. 

Bulk 

Weight 

Tolume 

Temp. 

itmos. excluded 

Fahr. 

per 

per 

cub. ft. 

lbs. pr. 

wat.=l 

Celsius 

inches 

lbs. pr. 

Scale. 

cub. ft. 

pound. 

per lb. 

cub. ft. 

at 39° 

Scale. 

mercury 

sq. in. 

T 

hf 

h 

c 

w 

V 

t 

i 

P 

101.36 

4301 

69.430 

.01617 

61.848 

1.0071 

30.83 

— 28.52 

— 14 

126.21 

6631 

94.369 

.01624 

61.583 

1.0130 

41.87 

— 26.48 

— 13 

111.67 

6583 

109.91 

.01639 

61.317 

1.0174 

48.74 

— 24.44 

— 12 

153.27 

7331 

121.58 

.01637 

61.101 

1.0210 

53.90 

— 22.41 

— 11 

162.51 

7974 

130.89 

.01638 

60.920 

1.0241 

58.00 • 

— 20.37 

— 10 

170.25 

8421 

138.69 

.01644 

60.7 b2 

1.0267 

61.44 

— 18.33 

— 9 

176.97 

SS12 

145.46 

.01617 

60.657 

1.0288 

64.43 

— 16.29 

— 8 

182.96 

9203 

151.5? 

.01652 

60.514 

1.0309 

67.09 

— 14.26 

— 7 

188.36 

9531 

156.97 

.01656 

60.372 

1.0333 

69.49 

—12.22 

— 6 

193.20 

9755 

161.87 

.01659 

60.282 

1.0359 

71.64 

— 10.18 

— 5 

197.60 

9975 

166.32 

.01663 

60.169 

1.0369 

73.60 

— 8.149 

— 4 

201.90 

10483 

170.67 

.01666 

60.072 

1.0385 

75.51 

— 6.111 

— 3 

205.77 

10398 

174.59 

.01669 

59.973 

1.0401 

77.23 

— 4.074 

— 2 

209.55 

10613 

178.42 

.01672 

59.896 

1.0416 

78.91 

— 2.037 

— 1 

212.00 

10824 

180.9 

.01674 

59.838 

1.0426 

100.00 

0.0000 

0 

213.04 

10883 

181.95 

.01675 

59.814 

1.0 430 

100.58 

0.6365 

0.3125 

216.33 

11047 

185.29 

.01677 

59.735 

1.0444 

102.45 

+ 2.037 

+ 1 

219.45 

11225 

138.45 

.01679 

59.659 

1.0 457 

104.36 

+ 4.074 

+ 2 

222.40 

11389 

191.44 

.01680 

59.592 

1.0469 

105.78 

+ 6.111 

+ 3 

225.25 

11550 

194.33 

.0168 L 

5.9.523 

1.0481 

107.35 

+ 8.149 

+ 4 

227.95 

11718 

197.08 

.01684 

59.459 

1.0492 

108.86 

+10.18 

+ 5 . 

230.60 

11868 

199.77 

.01686 

59.389 

1.0503 

110.33 

+ 12.22 

+ 6 

233.10 

12012 

202.40 

.01688 

59.329 

1.0514 

111.50 

+ 14.26 

+ 7 

235.49 

12150 

204.73 

.01690 

59.270 

1.0524 

113.05 

+ 16.29 

+ 8 

237.81 

122S2 

207.10 

.01692 

59.212 

1.0534 

114.00 

+ 18.33 

+ 9 

240.07 

12408 

209.39 

.01693 

59.154 

1.0545 

115.59 

+ 29.37 

+ 10 

242.24 

12528 

211.57 

.01695 

59.097 

1.0555 

116.«0 

+ 22.41 

+ 11 

244.32 

12642 

213.72 

.016.96 

59.057 

1.0564 

117.95 

+ 24.44 

+ 12 

246.35 

12750 

215.7S 

.01697 

59.01)6 

1.0573 

119.08 

+ 26.48 

+ 13 

248.33 

12852 

217.80 

.01698 

58.953 

1.0589 

120.18 

+ 28.52 

+ 11 

270.26 

12946 

219.76 

.01699 

5S.901 

1.0590 

121.25 

+ 30.55 

+ 15 

252.13 

13053 

221.67 

.01700 

58.851 

1.0599 

122.29 

+ 32.59 

+ 16 

253.98 

13157 

223.55 

.01701 

58.803 

1.0607 

123.32 

+ 34.63 

+ 17 

25577 

13258 

225.88 

.01702 

58.757 

1.0615 

124.32 

+ 30.67 

+ 18 

257.52 

13336 

227.16 

.01703 

58.713 

1.0623 

125.29 

+ 38.71 

+ 19 

259.22 

13430 

228.89 

.04704 

58.671 

1.0631 

126.23 

+ 40.74 

+ 20 

260.88 

13520 

230.59 

.01705 

58.631 

1.0639 

127.15 

+ 42.78 

+ 21 

262.50 

13608 

232.24 

.01707 

58.592 

1.0646 

128.05 

+ 44.82 

+ 22 

264.09 

13694 

233.86 

.01708 

58.560 

1.0654 

128.94 

+ 40.85 

+ 23 

265.65 

13778 

235.45 

.01709 

58.517 

1.0661 

129.80 

+ 48.89 

+ 24 

267.17 

13860 

237.00 

.01740 

58.481 

1.0668 

130.65 

+ 50.93 

+ 25 

268.66 

13940 

238.52 

.01711 

58.435 

1.0675 

131.48 

+ 52.97 

+ 26 

270.12 

14018 

240.02 

.01712 

58.400 

1.0684 

132.29 

+ 55.00 

+ 27 

271.55 

14094 

241.48 

.01713 

58.366 

1.0688 

133.05 

+ 57.04 

+ 28 

272.96 

14168 

242.92 

.OL714 

58.332 

1.0695 

133.86 

+ 59.08 

+ 29 

274.33 

14241 

244.32 

.01715 

58.298 

1.0701 

134.63 

+ 61.11 

+ 30 

275.68 

14314 

245.70 

.01716 

58.264 

1.0708 

135.38 

+ 63.15 

+ 31 

277.01 

14385 

247.06 

.01717 

58.230 

1.0714 

136.12 

+ 65.19 

+ 32 

278.32 

14454 

248.40 

.01718 

58.197 

1.0720 

136.84 

+ 67.23 

+ 33 

279.62 

14522 

249.73 

.01719 

58.164 

1.0726 

137.56 

+ 69.20 

+ 34 

280.89 

14592 

251.03 

.01720 

58131 

1.0732 

138.27 

+ 71.30 

+ 35 

282.14 

14659 

252.30 

.01721 

58.098 

1.0738 

138.96 

+ 73.34 

+ 36 

283.39 

14725 

253.58 

.01722 

58.066 

1.0744 

139.66 

+ 75.38 

+ 37 

284.58 

14789 

254.80 

.01723 

58.035 

1.0750 

140.33 

+ 77.41 

+ 38 

285.76 

14852 

256.01 

.01724 

58.004 

1.0756 

140.98 

+ 79.45 

+ 39 

286.96 

14913 

257.24 

.01725 

57.972 

1.0761 

141.64 

+ 81.49 

+ 40 

2-18.06 

14973 

25S.38 

.01726 

57.941 

1.0767 

142.27 

+ 83.52 

+ 41 

2 >9.2 4 

15032 

259.67 

.01727 

57.910 

1.0773 

142.91 

+ 85.56 

4- 42 

290.37 

15094 

260.71 

.01728 

57.879 

1.0778 

143.5 4 

-f* 8 / .^>1 

+ 43 

.91.48 

15149 

261.87 

.01729 

57.848 

1.0783 

144.15 

+ 89.64 

+ 44 

292.58 

' 15201 

262.99 

.01730 

57.817 

1.0789 

144.76 

+ 91.67 

+ 45 


i-.. 

















Properties of Water, 


899 





Water. 



Indie, press. 

Temp. 

Units of heat. 

Bulk 

Weight 

Volume 

Temp. 

Atmos, excluded 

Fair. 

per 

per 

cub. ft. 

lbs. pr. 

wat-.=l 

Celsius 

inches 

lbs. 

pr. 

in. 

Scale. 

CUD. ft 

pound. 

per lb. 

cub. ft. 

at 39° 

Scale. 

mercury 

sq. 

T 

h> 

h 

c 

w 

V 

t 

i 

V 

293.G6 

15265 

264.10 

.01731 

57.786 

1.0794 

145.37 

+ 93.71 

-J- 

46 

294.73 

15321 

265.20 

.01732 

57.769 

1.0799 

145.96 

-j- 95.75 

4" 

47 

29o.78 

15377 

266.27 

.01733 

57.742 

1.0804 

146.54 

-+ 97.78 

4- 

48 

296.82 

15432 

267.34 

.01734 

57.714 

1.0809 

147.12 

-f 99.82 

4- 

49 

297.84 

15485 

268.39 

.01735 

57.687 

1.0814 

147.69 

+ 101.8 

4- 

50 

298.85 

15536 

269.42 

.01735 

57.640 

1.0820 

148.25 

+ 103.9 

4- 

51 

299.85 

15588 

270.45 

.01736 

57.633 

1.0825 

148.80 

+ 105.9 

4“ 

52 

800.84 

15639 

271.46 

.01737 

57.606 

1.0S30 

149414 

+ 10^.0 

4- 

53 

301.81 

15690 

272.46 

.01737 

57.580 

1.0835 

149.89 

+ 110.0 

4- 

54 

302.77 

15739 

273.44 

.01738 

57.554 

1.0840 

150.43 

+ 112.0 

4" 

55 

308.72 

15789 

274.42 

.01739 

57.529 

1.0844 

150.95 

4- 114.1 

4- 

56 

804.69 

15839 

275.40 

.01739 

57.504 

1.0849 

151.48 

+ 116.1 

+ 

57 

305.60 

15888 

276.35 

.01740 

57.480 

1.0854 

152.00 

+ 118.1 

4- 

58 

806.52 

15936 

277.30 

.01741 

57.456 

1.0859 

152.51 

+ 120.2 

+ 

59 

307.42 

15983 

278.22 

.01741 

57.432 

1.0863 

153.01 

+ 122.2 

4- 

60 

308.38 

16029 

279.14 

.01742 

57.410 

1.0867 

153.51 

+ 124.3 

4- 

61 

309.22 

16075 

280.07 

.01743 

57.388 

1.0871 

154.01 

+ 126.3 

4- 

62 

310.11 

16120 

280.98 

.01743, 

57.364 

1.0875 

154.50 

+ 128.3 

4- 

63 

310.99 

16165 

281.87 

.01744 

57.344 

1.0880 

154.99 

4- 130.4 

4- 

64 

311.86 

16209 

282.78 

.01745 

57.322 

1.0884 

155.48 

4- 132.4 

4- 

65 

312.72 

16254 

283.66 

.01745 

57.300 

1.0888 

155.95 

4- 134.4 

+ 

66 

313.57 

16298 

284.54 

.01746 

57.278 

1.0892 

156.42 

4- 136.5 

4- 

67 

31442 

16342 

285.41 

.01746 

57.254 

1.0897 

156.90 

4-138.5 

+ 

68 

315.25 

16384 

286.27 

.01747 

57.232 

1.0901 

157.36 

4- 141.5 

4- 

69 

316.08 

16426 

287.12 

.01748 

57.210 

1.0905 

157.82 

4- 142.6 

4- 

70 

316.90 

16467 

287.96 

.01748 

57.188 

1.0909 

158.28 

4-144.6 

+ 

71 

317.71 

16507 

288.80 

.01749 

57.166 

1.0913 

158.73 

4- 146.7 

4- 

72 

318.51 

16547 

289.62 

.01750 

57.144 

1.0918 

159.17 

4- 148.7 

+ 

73 

319.31 

16587 

290.44 

.01751 

57.122 

1.0921 

159.62 

4-150.7 

+ 

74 

820.10 

16637 

291.26 

.01752 

57.101 

1.0926 

160.05 

4-152.8 

+ 

75 

320.88 

16677 

292.06 

.01752 

57.080 

1.0929 

160.49 

4-154.8 

4- 

76 

321.66 

16717 

292.85 

.01753 

57.059 

1.0935 

160.92 

4- 156.8 

+ 

77 

322.42 

16756 

293.65 

.01753 

57.038 

1.0937 

161.34 

4- 158.9 

+ 

78 

323.18 

16795 

294.43 

.01754 

57.017 

1.0941 

161.76 

4- 160.9 

4- 

79 

823.94 

16834 

295.21 

.01755 

56.996 

1.0945 

162.17 

4- 163.0 

4- 

80 

324.67 

16872 

295.96 

.01756 

56.975 

1.0949 

162.59 

4- 165.0 

4- 

81 

325.43 

16910 

296.75 

.01756 

56.954 

1.0953 

163.02 

4- 167.0 

+ 

82 

326.17 

16947 

297.51 

.01757 

56.933 

1.0956 

163.43 

4- 169.1 

4r 

83 

326.90 

16984 

298.26 

.01757 

56.912 

1.0960 

163.83 

4-171.1 

+ 

84 

327.63 

17020 

299.01 

.01758 

56.891 

1.0964 

164.24 

4-173.1 

4- 

85 

328.35 

17056 

299.75 

.01759 

56.871 

1.0968 

164.64 

4- 175.2 

4- 

86 

329.07 

17092 

300.50 

.01759 

56 862 

1.0972 

165.04 

4-177.2 

4- 

87 

329.78 

17127 

301.23 

.01760 

56.844 

1.0975 

165.43 

4-179.2 

4- 

88 

330.48 

17162 

301.95 

.01761 

56.826 

1.0979 

165.82 

4-181.3 

4- 

89 

331.18 

17197 

302.67 

.01761 

£6.808 

1.0982 

166.21 

4- 183.3 

4- 

90 

331.87 

17231 

303.38 

.01762 

56.790 

1.0986 

166.59 

4- 185.4 

+ 

91 

332.56 

17265 

304.10 

.01763 

56.772 

1.0989 

166.98 

4- 187.4 

4- 

92 

383.24 

17299 

304.80 

.01763 

56.754 

1.0993 

167.35 

4- 189.4 

+ 

93 

333.92 

17333 

305.50 

.01764 

56.735 

1.0996 

167.77 

4- 191.5 

4b 

94 

384 59 

17366 

306.19 

.01765 

56.716 

1.0999 

168.10 

4- 193.5 

+ 

95 

335.26 

17399 

306.88 

.01765 

56.699 

1.1003 

168.47 

4-195.5 

+ 

96 

330.58 

17465 

308.34 

.01767 

56.664 

1.1010 

169.21 

4-199.6 

4- 

98 

3 7.23 

17497 

308.91 

.01768 

56.647 

1.1013 

169 57 

4- 201.6 

+ 

99 

337.89 

17529 

309.60 

.01769 

56.629 

1.1017 

169.94 

4- 203.7 

4- 

100 

341.0 

17688 

312.87 

.01772 

56.549 

1.1035 

171.70 

4- 213.9 

+ 

105 

344.1 

17840 

316.04 

.01775 

56.469 

1.1050 

173.40 

+ 224.1 

+ 

HO 

347.1 

17993 

319.12 

.01778 

56.389 

1.1065 

175.06 

4- 234.2 

+ 

115 

350.0 

18136 

322.13 

.01781 

56.309 

1.1090 

176.68 

4- 244.4 

4- 

120 

352.8 

18278 

325.06 

.01784 

56.220 

1.1095 

178.25 

4-25F6 

+ 

125 

355.6 

18413 

327.91 

.01786 

56.146 

1.1100 

179.78 

4- 264.8 

-f 

130 

358.4 

1 18549 

330.75 

.01788 

56.073 

1.1124 

181.35 

4- 275.0 

j _J_ 

135 

-- 


































Properties of Steam. 


400 


Steam. 


Press 


Total pressure 

Temp. 

Volume 

Weight 

Bulk 

Units of heat, from 32° to T 

ob. at. 

lbs. pr. 

inches 

Falir, 

wat.-—1 

lbs. pr. 

cub. ft. 

Total pr. 

! Latent pr. 

lbs. pr. 

Sip. in. 

mer. 

Scale. 

at 39° 

cub. ft. 

pr. lb. 

pound. 

cub. ft. 

| pound. 

cub. ft. 

sip. in. 

P 

I 

T 

V 

\V 

C 

H 

11 

L 

U 

V 

1 

2.037 

101.36 

17983 

.00347 

288.24 

1112.8 

3.8614 

! 1043.4 

3.6337 

— 14 

2 

4.074 

126.21 

10353 

.00602 

165.94 

1120.4 

6.7449 

1026.0 

6.1165 

— 13 

3 

6.111 

141.67 

7283.8 

.09856 

116.75 

1125.1 

9.6308 

1015.2 

8.6901 

— 12 

4 

8.149 

153.27 

5608.4 

.01112 

89.895 

1128.7 

12.551 

1007.1 

11.199 

— 11 

5 

10.18 

162.51 

4565.6 

.01366 

73.180 

1131.5 

15.456 

1000.6 

13.714 

— 10 

6 

12.22 

170.25 

3851.0 

.01619 

61.742 

1133.8 

18.156 

995.17 

16.113 

— 9 

7 

14.26 

176.97 

3330.8 

.01872 

53.388 

1135.9 

20.846 

990.44 

18.194 

— 8 

8 

16.29 

182.96 

2935.1 

.02125 

47.046 

1137.7 

24.176 

936.22 

20.957 

— 7 

9 

18.33 

188.36 

2624.0 

.02377 

42.059 

1139.4 

27.083 

982.41 

23.352 

— 6 

10 

20.37 

193.20 

2373.0 

.02628 

38.037 

1140.8 

29.980 

978.99 

25.723 

— 5 

11 

22.41 

J 97.60 

2166.3 

.02880 

34.723 

1142.2 

32.895 

975.88 

28.099 

— 4 

12 

24.44 

201.90 

1993.0 

.03130 

31.945 

1143.5 

35.791 

972.84 

30.450 

— 3 

13 

26.48 

205.77 

1845.7 

.03380 

29.584 

1144.7 

38.691 

970.11 

32.789 

— 2 

14 

28.52 

209.55 

1718.9 

.03629 

27.551 

1145.8 

41.581 

967.43 

35.435 

— 1 

14.7 

29.92 

212.00 

1641.5 

.03800 

26.311 

1146.6 

43.571 

965.70 

36.706 

0 

15 

30.55 

213.04 

1608.6 

.03878 

25.784 

1146.9 

44.476 

964.96 

37.421 

0.3125 

] 6 

32.59 

216.33 

1511.7 

.04123 

24.230 

1147.9 

47.32S 

962.63 

39.690 

+ 1 

17 

34.63 

219.45 

1426.2 

.04374 

22.859 

1148.8 

50 248 

960.49 

42.012 

+ 2 

18 

36.67 

222.40 

1349.8 

.0 4622 

21.636 

1149.7 

53.138 

958.32 

44.393 

+ 3 

19 

38.71 

225.25 

1281.1 

.04868 

20.539 

1150.6 

56.011 

958,30 

46.698 

+ 4 

20 

40.74 

227.95 

1219.7 

.05119 

19.550 

1151.4 

58.894 

954.38 

48.655 

+ 5 

21 

42.73 

230.60 

1163.8 

.05369 

18.654 

1152.2 

61.758 

952.50 

51.924 

+ 6 

22 

44.82 

233.10 

1112.9 

.05605 

17.838 

1153.0 

64.637 

950.62 

53.282 

+ 7 

23 

46.85 

235.49 

1066.3 

.05851 

17.092 

1153.7 

67.503 

949.03 

55.529 

+ 8 

24 

48.89 

237.81 

1023.6 

.06095 

16.407 

1154.5 

70.367 

947.37 

57.743 

+ 9 

25 

50.93 

240.07 

984.23 

.06338 

15.776 

1155.1 

73.410 

945.76 

59.942 

lo 

26 

52.97 

242.24 

947.86 

.06582 

15.193 

1155.8 

76.074 

944.25 

62.161 

+11 

27 

55.00 

244.32 

914.14 

.06824 

14.652 

1156.4 

78.913 

942.74 

64.423 

+ 12 

28 

57.04 

246.35 

882.80 

.07067 

14.150 

1157.1 

81.772 

9 41.29 

66.521 

+ 13 

29 

59.08 

248.33 

853.60 

.07308 

13.682 

1157.7 

84.604 

933.88 

68.686 

+ Id 

30 

61.11 

250.26 

826.32 

.07550 

13.245 

1158.2 

87.444 

938.50 

70.857 

+ 15 

31 

63.15 

252.13 

800.79 

.07791 

12.835 

1158.8 

90.166 

937.17 

73.015 

+ 16 

32 

65.19 

253.98 

766.83 

.08031 

12.451 

1159.4 

93.121 

935.45 

75.126 

+ 17 

33 

67.23 

255.77 

754.31 

.08271 

12.090 

1159.9 

95.861 

934.57 

77.298 

+ 18 

34 

69.26 

257.52 

733.09 

.08510 

11.750 

1160.5 

98.782 

933.32 

79.425 

+ 19 

35 

71.30 

259.22 

713.08 

.08749 

11.429 

1161.0 

101.48 

932.10 

81.549 

+ 20 

36 

73.34 

260.88 

694.17 

.08987 

11.127 

1161.5 

104.38 

930.92 

83.662 

+ 21 

37 

75.38 

262.50 

676.27 

.09225 

10.840 

1162.0 

107.19 

929.76 

85.770 

+ 22 

38 

77.41 

264.09 

659.31 

.09462 

10.568 

1162.5 

109.98 

928.62 

87.866 

+ 23 

39 

79.45 

265.65 

643.21 

.09700 

10.310 

1162.9 

112.79 

927.51 

89.968 

+ 24 

40 

81.49 

267.17 

627.91 

.09936 

10.064 

1163.4 

115.59 

926.42 

92.059 

+ 25 

41 

83.52 

268.66 

613.34 

.10172 

9.S310 

1163.9 

118.39 

925.35 

94.126 

+ 26 

42 

85.56 

270.12 

599.46 

.10407 

9.60S6 

1164.3 

121.17 

924.30 

96.192 

+ 27 

43 

87.60 

271.55 

586.23 

.10642 

9.3963 

1164.7 

123.95 

923.28 

98.255 

+ 28 

44 

89.64 

272.96 

573.58 

.10877 

9.1938 

1165.2 

126.74 

922.27 

100.32 

+ 29 

45 

91.67 

274.33 

561.50 

.11111 

9.0002 

1165.6 

129.51 

921.29 

102.36 

+ 30 

46 

93.71 

275.68 

549.94 

.11344 

8.8149 

1166.0 

132.29 

920.32 

104.40 

+ 31 

47 

95.75 

277.01 

538.87 

.11577 

8.6374 

1166.4 

135.07 

919.36 

106.43 

+ 32 

48 

97.78 

278.32 

528.25 

.11810 

8.4673 

1166.8 

137.83 

918.43 

108.46 

+ 33 

49 

99.82 

279.62 

518.07 

.12042 

8.3040 

1167.2 

140.69 

917.49 

110.48 

+ 34 

50 

101.86 

280.89 

508.29 

.12273 

8.1472 

1167.6 

143.30 

916.58 

112.49 

+ 35 

51 

103.90 

282.14 

498.89 

.12505 

7.9966 i 

1167.9 

146.OS 

915.68 

114.50 

+ 36 

52 

105.93 

283.39 

489.85 

.12736 

7.8517 t 

1168.4 

] 48.«5 

914.79 

116.51 

+ 37 

53 

107.97 

284.58 

481.15 

.12966 

7.7122 

1168.7 

151.63 

913.93 

118.50 

+ 33 

54 

110.01 

285.76 

472.77 

.13196 

7.5779 

1169.0 

151.48 

913.08 

120.49 

+ 39 

55 

112.04 

286.96 

464.69 

.13428 

7.4468 \ 

1169.4 

157.02 

912.22 

122.47 

+ 40 

56 

114.08 

288.09 

456.90 

.13652 

7.3236 jj 

1169.8 

159.74 

911.42 

124.43 

+ 41 

57 

116.12 

289.24 

449.38 

.13883 

7.2030 

1170.1 

162.45 

910.48 

126.40 

+ 42 

58 

118.16 

290.37 

442.12 

.14111 

7.0866 j 

1170.5 

165.15 

939.78 

128.38 

+ 43 

59 

120.1H 

,291.48 

435.10 

.14338 

6.9741 

1170.8 

167.84 

908.97 

130.33 

+ 44 

60 

122.23 

292.58 

428.32 

.14566 

6.8654 

1171.2 

170.58 

908.18 

132.28 

+ 45 
































Properties op Steam. 


401 


Steam. 


Total pressure 

Temp. 

Volume 

Weight 

lbs. pr. 

inches 

Fahr. 

wat,=l 

lbs. pr. 

sq. in. 

mer. 

Scale. 

at 39° 

cub. ft. 

P 

I 

T 

V 

W 

61 

124.27 

293.66 

421.75 

.14792 

62 

126.30 

294.73 

415.40 

.15018 

63 

128.34 

295.78 

409.25 

.15244 

64 

130.38 

296.82 

403.29 

.15469 

65 

132.42 

297.84 

397.51 

.15694 

66 

134.45 

298.85 

391.90 

.15919 

67 

136.49 

299.85 

386.47 

.16130 

68 

138.53 

300.84 

381.18 

.16366 

69 

140.86 

301.81 

376.06 

.16590 

70 

142.60 

302.77 

371.07 

.16812 

71 

144.64 

303.72 

366.24 

.17035 

72 

146.68 

304.69 

361.53 

.17256 

73 

148.72 

305.60 

356.95 

.17478 

74 

150.75 

306.52 

352.49 

.17690 

75 

152.79 

307.42 

348.15 

.17919 

76 

154.83 

308.32 

343.93 

.18139 

77 

156.86 

309.22 

339.81 

.18359 

78 

158.90 

310.11 

335 80 

.18578 

79 

160.94 

310.99 

331.89 

.18797 

80 

162.98 

311.86 

328.08 

.19015 

81 

165.01 

312.72 

324.37 

.19233 

82 

167.05 

313.57 

320.74 

.19451 

83 

169.09 

314.42 

317.20 

.19668 

84 

171.12 

315.25 

313.74 

.19885 

S5r 

173.16 

316.08 

310.36 

.20101 

86 

175.20 

316.90 

307.07 

.20317 

87 

177.24 

317.71 

303.85 

.20532 

88 

179.27 

318.51 

300.70 

.20747 

89 

181.31 

319.31 

297.62 

.20962 

90 

183.35 

320.10 

294.61 

.21185 

91 

185.38 

320.88 

291.66 

.21390 

92 

187.42 

321.66 

288.78 

.21603 

93 

189.46 

322.42 

285.96 

.21816 

94 

191.50 

323.18 

283.21 

.22029 

95 

193.53 

323.94 

280.50 

.22241 

96 

195.57 

324.67 

277.86 

.22453 

97 

197.61 

325.43 

275.27 

.22672 

98 

199.65 

326.17 

272.73 

.22875 

99 

201.68 

326.90 

270.24 

.23085 

100 

203.72 

327.63 

267.80 

.23296 

101 

205.76 

328.35 

265.41 

.23505 

102 

207.79 

329.07 

263.07 

.23715 

103 

209.83 

329.78 

260.77 

.23924 

104 

211.87 

330.48 

258.52 

.24132 

105 

213.91 

331.18 

256.31 

.24340 

106 

215.94 

331.87 

251.14 

.24548 

107 

217.98 

332.56 

252.01 

.24750 

108 

220.02 

333.24 

249.92 

.24963 

109 

222.05 

333.92 

247.87 

.25169 

110 

224.10 

334.59 

245.86 

.25375 

111 

226.13 

335.26 

243.88 

.25581 

113 

230.20 

336.58 

240.03 

.25991 

114 

232.24 

337.23 

238.15 

.26204 

115 

234.28 

337.89 

236.31 

.26400 

120 

244.4 

341.0 

227.56 

.27421 

125 

254.6 

344.1 

219.50 

.28422 

130 

264.8 

347.1 

212.07 

.29419 

135 

275.0 

350.0 

205.18 

.80406 

140 

285.2 

352,8 

198.78 

.31385 

145 

295.4 

355.6 

192.83 

.32354 

150 

305.6 

358.4 

187.26 

.33315 


Bulk 

Units of heat, 

from 32° to T 

cub. ft. 

Total pr. 

Latent pr. 

pr. lb. 

pound. 

cub. ft. 

pound. 

cub. ft. 

<7 

H 

IT 

L 

L' 

6.7601 

1171.5 

173.27 

907.40 

134.22 

6.6583 

1171.8 

175.96 

906.63 

136.16 

6.5597 

1172.1 

178.65 

905.87 

138.09 

6.4642 

1172.5 

181.34 

905.13 

140.01 

6.3715 

1172.8 

184.03 

904.39 

141.93 

6.2817 

1173.1 

186.72 

903.66 

143 85 

6.1994 

1173.4 

189.40 

902.94 

145.64 

6.1099 

1173.7 

192.07 

902.23 

147.66 

6.0277 

1174.0 

194.74 

901.53 

149.56 

5.9478 

1174.3 

197.42 

900 84 

151.45 

5.8702 

1174.6 

200.08 

900.15 

153.34 

5.7948 

1174.9 

202.74 

899.46 

155.21 

5.7214 

1175.1 

205.40 

898.79 

157.09 

5.6500 

1175.4 

208.04 

898.13 

158.88 

5.5805 

1175.8 

210.67 

897.57 

160.83 

5.5129 

1176.0 

213.30 

896.83 

162.67 

5.446S 

1176.2 

215.93 

896.18 

164.56 

5.3825 

1176.5 

218.56 

895.54 

166.37 

5.3190 

1176.8 

221.19 

894.92 

168.22 

5.2588 

1177.0 

223.82 

894.27 

170.04 

5.1992 

1177.3 

226.44 

893.65 

171.87 

5.1410 

1177.6 

229.06 

893.03 

173.70 

5.0843 

1177.9 

231.68 

892.51 

175.52 

5.0289 

1178.1 

234.28 

891.82 

177.33 

4.9748 

1178.3 

236.89 

891.22 

179.14 

4.9219 

1178.6 

239.50 

890.63 

180.95 

4.8703 

1178.8 

242.10 

890.04 

182.75 

4.8198 

1179.1 

244.69 

889.46 

184.53 

4.7704 

1179.3 

247.29 

888.8S 

186.33 

4.7222 

1179.6 

249.88 

888.31 

188.12 

4.6750 

1179.8 

252.45 

887.74 

189.88 

4.6288 

1180.0 

255.02 

887.19 

191.66 

4.5836 

1180.3 

257.58 

886.63 

193.43 

4.5394 

1180.5 

260.14 

886.08 

195.19 

4.4961 

1180.7 

262.69 

885.53 

196.94 

4.4537 

1180.9 

265.23 

885.00 

198.71 

4.4106 

1181.2 

267.77 

884.45 

200 49 

4.3715 

1181.4 

270.30 

883.91 

202.18 

4.3316 

1181.6 

273.10 

883.38 

203.92 

4.2926 

1181.9 

275.52 

882.85 

205.67 

4.2543 

1182.1 

277.85 

882.33 

207 39 

4.2167 

1182.3 

280.38 

881.81 

209.12 

4.1799 

1182.5 

282.90 

881.29 

210 84 

4.1438 

1182.7 

285.42 

880.78 

212.55 

4.1083 

1182.9 

287.93 

880.27 

214.26 

4.0736 

1183.2. 

290.45 

879.77 

215.96 

4.0394 

1183.4 

292.94 

879.27 

217.66 

4.0058 

1183.6 

295.41 

879.79 

219.36 

3.9731 

1183.8 

297.91 

878.28 

221.05 

3.9408 

1183.9 

300.44 

877.80 

222.74 

3.9091 

1184.2 

302.93 

877.31 

224.42 

3.8474 

1184.6 

307.90 

876.25 

227.74 

3.8100 

1184.8 

310.36 

875.88 

229.51 

3.7878 

1185.0 

312.86 

875.40 

231.10 

3.6475 

1185.9 

325.20 

873.09 

239.41 

3.5184 

1186.9 

337.39 

870.85 

247.51 

3.0991 

1187.8 

319.44 

868.68 

255.55 

3.2880 

1188.7 

361.42 

866.56 

263.48 

3.1862 

1189.5 

373.34 

864.49 

271.32 

3.0908 

1190.4 

385.20 

862.48 

278.97 

3.0001 

1191.2 

396.86 

860.45 

286.66 


Press 
ob. at. 
lbs. pr. 
sq. in. 


+ 46 
+ 47 
+ 48 
+ 49 
+ 50 
+ 51 
+ 52 
+ 58 
+ 54 
+ 55 
+ 56 
+ 57 
-f 58 
+ 59 
+ 60 
+ 61 
+ 62 
+ 63 
+ 64 
+ 65 
+ 66 
+ 67 
-f~ 68 
+ 69 
+ 70 
+ 71 
+ 72 
+ 73 
+ 74 
+ 75 
+ 76 
+ 77 
+ 78 
+ 79 
+ 80 
+ 81 
+ 82 
+ 83 
+ 84 
+ 85 
+ 86 
+ 87 
+ 88 
+ 89 
+ 90 
+ 91 
+ 92 
+ 93 
+ 94 
+ 95 
+ 96 
+ 98 
+ 99 
+ 100 
+ 105 
+ 110 
+-115 
4-120 
4-125 
4-130 
+ 135 


26 


















102 


Expansion of Steam. 



EXPANSION OP STEAM. 

In order to save steam, or-smore correctly to employ its effect to a higher 
degree, the admittance of steam to the cylinder is shut off when the piston 
has moved a part of the stroke ; from the cut-off point the steam acts ex¬ 
pansively with a decreased pressure on the piston, as represented by the 
accompanying figure. 

Let the steam be cut off at ? of the stroke, and 
Aa represent the total pressure, say 20 pounds 
per square inch which will continue to the point 
E where the admittance of steam is shut off at 
one-third the stroke S. The steam Aa eE, is now 
acting expansively on the piston, and the pres¬ 
sure decreases as the volume increases, when the 
piston has attained Cc or two-thirds of S, the 
pressure C’c= 10 pounds, only half the pressure 
Aa= 20 because the volume Aa eE is only half of 
A a cC, and so on until the piston has attained B b 
the pressure B'b—}i X20=6-66 pounds. 

The mean pressure, or the effectual pressure, 
throughout the stroke, will be about 13-33 pounds 
per square inch, or 66 per cent., but the quantity 
of steam used is only 33 per cent., hence 33 per 
cent, is gained by using the steam expansively. 

I = part of the stroke S in feet, at which the steam is cut off. 

P— pressure per square inch under full admittance of steam. 

F=mean pressure per square inch throughout the stroke S. 
f — mean pressure per square inch during the expansion, which in 
double expansion cylinder engines will be the average pressure per 
square inch on the large piston A. , 

p = end pressure per square inch after expansion. 

S = stroke of the cylinder Piston in feet. 

. FS—Pl PI 

S— l ’ P ~ S' 

The following Tables are calculated from these formulas. 

Example 1. Required from the Table I. the mean pressure F for P=32 
lbs. at five-eights expansion. 

Add | 3 2 ° ^ s - } from the Table I. 

Mean pressure of 32 lbs. F=23-735 the answer. 

Example 2. Required from Table II. the mean pressure /, per square 
inch during the expansion, or on the large piston A in double cylinder 
engines, when the initial pressure P= 75 lbs. and under two-thirds ex¬ 
pansion 1 /=40-75 Table II. 

Example 3. Required the mean pressure /=! for an initial pressure 
P=43 lbs. under % expansion! 

For P = 40 lbs. /= 18-48 j TT 
P = 30 or 3 lbs. f= 1-38] Table II# 

P = 43 lbs. f = 19 86 the answer. 

The effect gained or fuel saved by expansion and high steam is calculated 
from the following formulae, in which it is supposed as a unit the work 
of an engine with P=30 pounds per square inch, or an indicated pressure 
of 15 lbs. without expansion. 

c = per cent on 100, of effect gained or fuel saved. 


■£ f '= : ^2.3(log. S- —log. Zj+lJ 


For expansion c= 100 (1- 


IP 

SF 


26490 

). For high steam c— 100 (1— ^ ). 


The following Table III. is calculated from these formulae, in which 
the first line from 30 contains the economy per cent, from expansion 
alone, and the column o contains the economy per cent, from high steam 
above P =30 lbs. The balance of the table contains tne jointed economy 
of expansion and high steam. Required the jointed economy of P=90 
lbs. under £ expansion! 60-5 per cent, the answer. 
























Expansion Table I. Mean Pressure F, 


403 


Grade of .Expansion* 


Press. 

P. 

1 

1 1 

, t 

1 i 

1 

1 2 

3 

1 

[ * 

1 

0*9637 0*9333 

0*9187 

0*8465 

0*7417 

' 0*6991 

0*5965 

' 0 3848 

2 

1*9275: 1*8686 

1*8375 

1*6930 

1*4835 

1*3982 

1*1930 

0*7697 

3 

2*8912 

2*7999 

2*7562 

2*5395 

2*2252 

2*0873 

1*7895 

1*1546 

4 

3*8550 

3*7333 

3*6750 

3*3860 

2*9670 

2*7964 

2-3860 

1*5395 

5 

4*8185 

4*6666 

4*5935 

4*2325 

3*7085 

3*4955 

2*9825 

1*9240 

6 

5*7822 

5*5999 

5*5 122 

5*0790 

4*4502 

4*1946 

3*5790 

2*2008 

7 

6*7469 

6*5334 

6*4319 

5*9255 

5*2419 

4*8937 

4*1755 

2*6946 

8 

7*7106 

7*5666 

7*3396 

6*7720 

i 5*9346 

5*5928 

4*7720 

3*0784 

9 

8*6733 

8*3999 

8*2683 

7*6185 

' 6*6753 

6*2919 

5*3685 

3*4632 

10 

9*6370 

9*3333 

9*1870 

8*4656 

| 7*4170 

6*9912 

5*9650 

3*8480 

11 

10*601 

10*266 

10*106 

9*3115 

8*1597 

1 7*7001 

6*5615 

4*2338 

12 

11*565 

11*199 

10*925 

10*158 

8*9014 

8*3892 

7*1580 

4*6186 

13 

12*528 

12*133 

11*943 

11*004 

9*6421 

9*0783 

7*7545 

5*0034 

14 

13*492 

13*066 

12*862 

11*851 

10*384 

9-7874 

8*5310 

5*3882 

15 

14*456 

13*930 

13*781 

12*698 

11*126 

10*486 

8*9475 

5*7730 

16 

15*420 

14*933 

14*700 

13*544 

11*268 

11-185 

9*5440 

6*1582 

17 

16*383 

15*866 

15*618 

14*390 

12*609' 

11*884 

10*140 

6*5426 

18 

17*347 

16*799 

16*537 

15*237 

13*531 

12*483 

10-737 

6*9274 

19 

18*311 

17*733 

17*448 

16*803 

14*093 

13*183 

11*333 

7*3122 

20 

19*275 

18*666 

18*375 

16*930 

14*835 

13*902 

11*930 

7*6970 

21 

20*238 

19*599 

19*293 

17*776 

15*576 

14*501 

12*526 

8*0818 

22 

21*201 

20*532 

20*211 

18*623 

16*318 

15*300 

13*123 

8*4667 

23 

22*166 

21*466 

21*131 

19*469 

17 060 

15*989 

13*720 

8*8516 

24 

23*030 

22*399 

22*050 

20*316 

17*802 

16*698 

14*316 

9*4365 

25 

| 24*093 

23*333 

22*968 

21*162 

18*573 

17*477 

14*912 

9*6210 

26 

25*057 

24*286 

23*887 

22*009 

19*285 

18*046 

15*509 

9*8978 

28 

26*985 

26*233 

25*714 

23*702 

20*769 

19*495 

16*702 

10*775 

30 

28*912 

27*999 

27*562 

25*395 

22*252 

20*873 

17*895 

11*546 

35 

33*731 

32*666 

32*156 

29*627 

25*961 

24*368 

20*877 

13*470 

40 

38*550 

37*333 

36*750 

33*860 

29*670 

27*964 

23*860 

15*395 

45 

43*368 

42*000 

41*341 

38*092 

33*378 

31*459 

26*842 

17*319 

50 

48*187 

46*666 

45*937 

42*325 

37*067 

34*955 

29*825 

19*243 

55 

53*005 

51*333 

50*530 

46*557 

40*775 

38*450 

32*807 

21*167 

60 

57*822 

55*999 

55*122 

50*790 

44*520 

41*946 

35*790 

23*090 

65 

62*640 

60*666 

59*715 

55*022 

48*228 

45*441 

38*772 

24*924 

70 

67*460 

65*333 

64*300 

59*255 

52*419 

48*937 

41*755 

26*694 

75 

72*278 

69*999 

68*893 

63*487 

56*127 

52*432 

44*737 

2 S -626 

80 

77*096 

75*666 

73*500 

67*720 

59*340 

55*928 

47*720 

30*790 

85 

81*914 

80*333 

78*093 

71*952 

63*048 

59*423 

50*702 

32*714 

90 

86*730 

83*999 

82*680 

76*180 

66*750 

62*919 

53*680 

34*638 

95 

91*548 

8 S -666 

87*273 

80*412 

70*458 

66*414 

56*662 

36*554 

100 

96*370 

93*333 

91*870 

84*650 

74*170 

69*910 

59*650 

38*480 

105 

101*18 

97*999 

96*463 

88*882 

77*878 

73*405 

62*632 

40*404 

110 

105*99 

101*66 

101*05 

93*120 

81*586 

76*900 

66*614 

42*328 

115 

110*80 

106*33 

105*64 

97*352 

85*294 

80*395 

69*596 

44*252 

125 

120*46 

115*66 

114*83 

105*81 

102*83 

87*387 

74*562 

48*10] 

140 

134*92 

130*66 

12 S -62 

118*51 

103 84 

97*874 

85*310 

53*882 

150 

14.*56 

139*33 

137*81 

126*47 

111*26 

104*86 

89*470 

57*730 

200 

192*75 

186*66 

183*75 

169*30 

148*35 

139*02 

119*30 

76*970 

250 

240*93 

233*33 

229*68 

211*62 

185*43 

174*77 

149*12 

96*210 

300 

289*12 

279*99 

275*62 

253*95 

222*52 

208*73 

178*95 

115*46 





















































404 


Expansion Table IT. for Double Cylinder Expansion Engines. 


Mean Pressure f during the Expansion, 


Pr^s. 

P. 

1 * 

1 1 

3 

3 

8 

4 

1 $ 

1 2 

3 

1 * 

| i 

30 

28-549 

24 

23-50 

20-79 

17-60 

16-31 

j 13-86 

1 8-9097 

35 

33-308 

28 

27-41 

24-25 

20-54 

19-02 

16-17 

10-394 

40 

38-066 

32 

31-83 

27-72 

23-47 

21-73 

18-48 

| 11-879 

45 

42-824 

36 

35-25 

31-18 

26-40 

24-46 

20-79 

, 13-364 

50 

47-582 

40 

39-16 

34-65 

29-33 

27-16 

23-10 

14-849 

55 

52-340 

44 

43-08 

38-11 

32-24 

30-17 

25-41 

16-334 

60 

57-098 

48 

47-00 

41-58 

35-20 

32-62 

27-72 

17-819 

65 

61-853 

52 

50-91 

45-04 

38-14 

35-33 

30-03 

19-303 

70 

66-616 

56 

54-83 

48-51 

41-07 

38-04 

32-34 

20-788 

75 

71-371 

60 

58-75 

51-90 

44-00 

40-75 

34-65 

22-263 

80 

76-128 

64 

62-66 

55-44 

46-94 

43-47 

36-96 

23-758 

85 

80-885 

68 

66-18 

58-90 

49-87 

46-19 

39-27 

25-243 

90 

86-448 

72 

70-50 

62-37 

52-80 

48-93 

41-58 

26-729 

95 

90-391 

76 

74-41 

65-73 

55-73 

51-62 

43-89 

28-213 

100 

95-160 

80 

78-33 

69-30 

58-66 

54-33 

46-20 

29-699 

105 

99-910 

84 

82-24 

72-76 

61-57 

57-33 

48-51 

31-183 

1L0 

104-68 

88 

86-46 

76-23 

64-48 

60-35 

50-82 

32-669 

115 

109-40 

92 

90-08 

79-69 

67-44 

62-79 

53-13 

34 153 

125 

118-95 

100 

97-91 

97-02 

73-34 

67-95 

57-75 

37-122 

140 

133-23 

112 

109-6 

97-02 

82-14 

76-08 

64-68 

41-576 

150 

142-74 

120 

117-5 

103.9 

88-00 

81-50 

69-30 

44-548 

200 

190-32 

160 

156-6 

138.6 

117-3 

108-6 

92-40 

59-398 

250 

237-07 

200 

195-7 

173.2 

146-6 

135.8 

115-5 

74-247 

300 

288-16 

240 ! 

235-0 

207.9 

176-0 

163.1 

138-6 

89-097 


Table III. Economy of Expansion and high Steam. 
Fuel saved or effect gained per cent. 


Pres . 

P. | 

0 

* 

i | 

3 

8 

i 

t § 

1 

i 

30 

0 

12 

29-5 

32 

41 

49-3 

52 

58 

67-5 

35 

1-6 

13-6 

31 

33-6 

42-6 

51 

53-6 

59-6 

69-1 

40 

2-5 

14-5 

32 

34-5 

43-5 

51-8 

54-5 

60-5 

70 

45 

3-4 

15-4 

33 

35-4 

44-4 

52-7 

55-4 

61-4 

71 

50 

4.3 

16-3 

33-8 

36-3 

45-3 

53-6 

56-3 

62-3 

71-8 

55 

5-2 

17-2 

34-7 

37-2 

46-2 

54-5 

57-2 

63-2 

72-7 

60 

6 

18 

35-7 

38 

47 - 

55-3 

58 

64 

73-5 

65 

6-7 

18-7 

36.2 

38-7 

47-7 

56 

58-7 

64-7 

74-2 

TO 

7-3 

19.3 

36.8 

39-3 

48-3 

56-6 

59-3 

65-3 

74-8 

75 

7-8 

19-8 

37-3 

39-8 

48-8 

57-1 

59-8 

65-8 

75*3 

80 

8-5 

20-5 

38 

40-5 

49-5 

57-8 

60-5 

66-5 

76 

85 

9 

21 

38-5 

41 

50 

58-3 

61 

67 

76-5 

90 

9-5 

21-5 

39 

41-5 

50-5 

58-8 

61-5 

67-5 

77 

95 

10 

22 

39-5 

42 

51 

59-3 

62 

68 

77-5 

100 

' 10-4 

22-4 

40 

42.4 

51.4 

59-7 

62-4 

68-4 

78 

105 

! 10-7 

22-7 

40-2 

42-7 

51-7 

60 - 

62-7 

68-7 

78-2 

115 ! 

11 

23 

40-5 

43 

52 

60-3 

63 

69 

78-5 

125 ! 

11-7 

23-7 

41-2 

43-7 

52-7 

61 

63-7 

69-7 

79-2 

150! 

14 

26 

43-5 

46 

55 

63-3 

66 

72 

81-5 

200 

16 

28 

45-5 

48 

57 

65-3 

68 

74 

83-5 

250 

17-7 

29-7 

46-2 

49-7 

58-7 

67 

69-7 

75-7 

85-2 

300 

19 

31 

48-5 

51 

60 

i 

68-3 

71 

77 

86.5 


































































CONSCMPTION OF FUEL 


toa 


Table IV. 

Consumption of Coal in pounds per horse power per hour. 


Pres. 


1 4 

1 

1* 3 

l 

5 

2 

3 

1 I 

P. 

0 

1 4 

3 

8 

2 

8 

3 

4 


lbs. 

lbs, 

lbs, 

lbs, 

lbs, 

lbs, 

lbs, 

lbs, 

lbs, 

lbs, 

30 

5-6 

4-93 

3*95 

3*81 

3*30 

2*84 

2*69 

2*35 

1*82 

35 

5-5 L 

4-84 

3*86 

3 72 

3*21 

2*74 

2*60 

2*26 

1*73 

40 

5-46 

4-79 

3*81 

3*67 

3*16 

2-70 

2* 55 

2*21 

1*68 

45 

5-41 

4-73 

3*75 

3*62 

3*11 

2-65 

2*50 

2*16 

1*62 

50 

5*36 

4-68 

3*71 

3*57 

3*06 

2*60 

2*45 

2*11 

1*58 

55 

5*31 

4-63 

3*66 

3*51 

3*01 

2*55 

2*40 

2*06 

1*53 

60 

5*26 

4-59 

3*60 

3*47 

2*97 

2*50 

2*35 

2*02 

1*49 

65 

5-20 

4-55 

3*57 

3*43 

2*93 

2*46 

2*31 

1 *98 

1*45 

70 

5*19 

4*52 

3*54 

3*40 

2*90 

2*43 

2*28 

1*94 

1*41 

75 

5-16 

4.49 

3*51 

3*37 

2-87 

2*40 

2*25 

1*91 

1*39 

80 

5-12 

4-45 

3*47 

3*33 

2*83 

2*36 

2*21 

1*88 

1*35 

85 

5-09 

4-42 

3*44 

3*30 

2-80 

2-33 

2*18 

1*85 

1*32 

90 

5-07 

4*39 

3*41 

3*28 

2*77 

2*31 

2*16 

1*82 

1*29 

95 

5-04 

4*37 

3-39 

3*25 

2*74 

2*28 

2*13 

1 *79 

1*26 

100 

5-01 

4*34 

3*36 

3*23 

2*72 

2*26 

2 10 

1*77 

1*23 

105 

5-00 

4*32 

3*35 

3*21 

2*70 

2*24 

2*09 

1*75 

1*22 

115 

4-98 

4*31 

3*33 

319 

2*69 

2*22 

2*07 

1*73 

1*20 

125 

4-94 

4*27 

3*29 

3*15 

2*65 

2*19 

2*03 

1*70 

1*17 

150 

4-81 

4*14 

3*16 

3*02 

2*52 

2*05 

1*90 

1*57 

1*04 

200 

4-70 

403 

3*05 

2*91 

2 41 

1*94 

1*79 

1*46 

0*92 

250 

4-60 

3-93 

3*01 

2*81 

2*31 

1*85 

1*70 

1*36 

0*83 

300 

4-54 

3*87 

2*89 

2*75 

2*24 

1*78 

1*62 

1*29 

0*75 


Table V. 

Consumption of Coal in tons per 100 horses in 24 hours. 


Pres. 

P. 

0 

1 * 

1 £ 

3 

8 


i 

i 

1 

i 

lbs. 1 

30, 

tons! tons, 

6*00 5*29 

tons, 

4*23 

tons, 

4*09 

tons, 

3*54 

tons, 

3*04 

tons, 

2*88 

tons, 

2*52 

tons, 

1*95 

35! 

5*90 

5*19 

4*13 

3*99 

3*44 

2*94 

2-79 

2*42 

1*86 

40 

5-85 

513 

408 

3*93 

3*39 

2*90 

2-73 

2 37 

1*80 

45 

5*80 

5*07 

4*02 

3*88 

3*34 

2*84 

2*68 

2*31 

1*73 

50 

5*75 

5 01 

3*97 

3*83 

3*28 

2-79 

2*63 

2*26 

1*69 

55 

5*70 

4*96 

3*92 

3*77 

3*22 

2*73 

2*57 

2*21 

1*64 

60 

5*64 

4*92 

3*87 

3*72 

3*18 

2*68 

2*52 

2-17 

1*60 

65 

5*58 

4*88 

3*82 

3*68 

3*14 

2*63 

2*48 

2*12 

1*55 

70 

5*56 

4*84 

3*79 

3*64 

3*11 

2*60 

2*44 

2*08 

1*51 

75 

5*53 

481 

3*76 

3*61 

3*07 

2-57 

2*41 

2*05 

1*49 

80 

5*19 

4*77 

3*72 

3-57 

3*03 

2*53 

2*37 

2*01 

1-44 

85| 

5*46 

4*74 

3 69 

3*54 

3*00 

2*50 

2*33 

1*98 

1*41 

90 

5*43 

4*70 

3*66 

3*51 

3-97 

2-47 

2*31 

1*95 

1*38 

95 1 , 

5*40 

4*68 

3*63 

3*48 

2*94 

2*44 

2*28 

1*92 

1*35 

1001 

5*37 

4*65 

3*60 

3*46 

2*91 

2* 42 

2*26 

1-90 

1*32 

105| 

5*36 

4-63 

3*59 

3*44 

2*89 

2*40 

2*24 

1*88 

1-31 

115| 

5*34 

4*61 

3*57 

3*42 

2*88 

2-38 

2*22 

1*85 

1*29 

125 

5*30 

4*58 

3*53 

3*38 

2-84 

2*34 

218 

1*82 

1*25 

150 

5*16 

4*44 

3.39 

3*34 

2*81 

2-30 

2*04 

1*68 

1*11 

200, 

5*04 

4*32 

3*27 

312 

2*59 

219 

1*92 

1*56 

0 99 

250 

4*93 

4*21 

3*22 

3*01 

2*47 

2*09 

1*82 

1*46 

0*89 

300 

4*87 

4*15 

3*10 

2 95 

2*40 

2*01 

1*74 

1 38 0*83 

























































4)3 


Force and Air Pumps. 


Force or Feed. Pumps. 

Letters denote , 

s = stroke^ 1 } of the force-pump, single acting. 

^~stroke tei } ^ ie s t eam ‘ c yli n der piston, in inches, double acting, 

F= volume of steam given in the table at the given pressure. 

The stroke of the steam-piston is only that under which steam is fully admitted 
to the cylinder. 


d 


-2 dJs 

\ Fs 


Vd 

Slip-water included in the formulas. 


4,5. 


Example. Required, the diameter of a force-pump having tire same stroke as the 
cylinder piston s = 38 inches, diameter of cylinder D = 30 inches. The steam is cut 
off at £ the stroke, and the steam pressure -f 50 pounds per square inch. Here V 
— 437, and £= 19 inches, because steam is cut off at i the stroke. 


d = 2 X 30 


V* 


19 


= 2.03 inches. 


1 437 X 38 

To find tlie Quantity of Condensing Water. 


g = condensing water of temp, t in cubic feet. 
Q — steam of temperature T in cubic feet. 

V — temperature in the condenser. 


1.4Q(990 -p T — V\ 
q ~ n t'-t) ’ 


Dimensions of tlie Air-Pump. 

d = 2.3 ! S ( 990 + . 

\ Vs(t'-t) 


d = diameter \ of the air-pump, 
s = stroke j single acting. 

2) = diameter ) of the steam cylinder, 

£= stroke j double acting. 

Assume t' = 100°, and t = 50°, we shall have 
Single acting air-pumps. 

d= 0.326X>. 


7. 




s =0.1062)2 


£(940 + T) 
Vs 
£(940 + T) 
Vd* ’ 


8 . 


9. 


Double acting air-pumps. 
: = 0.23 


s = 0.0532)2 


[ £(940 + T) 
Vs 
£(940 + T) 
Vd* ’ 


10 . 


11 . 


Example. A single acting air-pump is to be constructed for an engine D = 
38 inches, £=45 inches stroke of the cylinder; the stroke of the air-pump can be 
32 inches, and the exhaust steam is 261°. Required, the diameter of the air- 
pump 1 V= 767. 


<2 = 0.326X38 


4 


45(940 + 261) 
767 X 32 


18.25 inches. 


Slip-water included. Tand F must be taken for the exhaust steam, as the 
steam may have worked expansively; the area of the foot valve must be calcu¬ 
lated from the following formulas. 

Foot Valve in tlie Air-Pump. 

To enable an air-pump to work well and with the greatest advantage, it is neces¬ 
sary to pay particular attention to the following formulas. The force by which 
the water is driven from the condenser through the foot valve into the air-pump 
is limited by the pressure in the condenser; this pressure is the vacuum sub¬ 
tracted from 14.7 pounds; it is noted in the third column, where the temperature 
in the condenser is opposite, in the first column. Every pound of this pressure 
per square inch balances a column of water 27 inches high, which is the head that 
presses the water from the condenser. 



















Air-Pump. 


407 

.) 


Foot-Valves In Air-Pumps. 

= area of the air-pump piston. a = ^ n 

a = area of the foot-valve, or bucket-valve. - 23000 mV V V 

333 = diameter of the air-pump-piston. w = 0-6 to 0‘8 

tl = diameter of the foot-valve, when round. 


<S = stroke of air-pump piston, in feet. 

33 = pressure in the condenser at the temperature T. 

» = number of strokes of the air-pump piston per minute. 


II 

o ® 

< w» 

«i * 

12, 

is V s « 

tJ = -=, 

10 Vi 

15, 

iooa Vi 

13, 

s _ioowi 

n 33 a ’ 

16, 

_ 100a Vi 
•as ’ 

14, 

„ ioowi 

17. 


Example. The area of an air-pump-piston is 2 - 35 square feet, stroke of 
piston = 3-6 feet, to make n =. 40 strokes per minute, and the pressure to be 
Jjj) = 3-2 pounds. Required the area of the foot-valve. 


2-35X3-6X40 

100 ^ 2 “ 


1-85 square feet. 


To Find, tlie Velocity and Quantity of the Injection Water 
through the Adjustage into the Condenser# 

Letters denote. 


v = velocity in feet per second. 

h = head of the press water; •+■ when above, and — below the adjustage. 

F— vacuum, noted — or negative in the last column, but is positive in th® 
formulas. 

q = quantity of water discharged in cubic feet, per second. 
a — area of all the holes in the adjustage in square feet. 

L == length^*} i n 5 ec ti°n pipe, in feet. 

» — double strokes of cylinder-piston, or revolutions per minute. 

A, D, and S, dimensions of the steam cylinder, in feet. 

T — temperature, and v e= volume coefficient of the exhaust st6am. 


a = 


5 V '2 F±h’ 


v =» 8 V 2F+h 
D*(94 


55 V 


18, 

q — 5a V2 F+h, . 

19, 

- °- 35 </iS’ 

on 

n S D‘ i (940+T) 

AU, 

a " 275FV 2 F+h 


21 

32, 

23 , 






























BTEAU. 


1^9 


Example. Required the diameter of an injection pipe L = 10 feet long, 
which shall supply q — 1-3 cubic feet of water per second into a vacuum of 12 
pounds per square inch, the head of press water h — 2 feet ? 


d = 0-35 


10X1-3 

2X12+2 


0*3055 feet = 34-4 inches. 
J b 


Area of Steam Passages* 

a = area of the steam pipe, sq. in. 

A = area of the cylinder piston, sq. in. 
d = diameter of the pipe, in inches. 

D — diameter, S — stroke of cylinder, in inches. 


_ ASn 
a 35000’ 


d = vysn 
186 


2-1, 25. 


Example. Required the diameter of a steam-pipe for a cylinder D = 40 
inches. Stroke of piston S = 48 inches, and n = 38 revolutions per minute? 


d = fOiAtS^ gg _ Q . 2 inchegj near]y> 
186 

Steam Ports to the Cylinder* 

A Sn 


a = 


30600’ 


26, 


Safety Valve# 

Three-fourths of the fire grate in square feet is a good proportion for the 
safety valve in square inches. 


Notation of Letters corresponds with Figure 3, Plate V. 
a = area of safety valve in square inches. 
p= pressure per square inch in the boiler 'l 
W= weight on the safety valve lever >in pounds. 

Q = weight of the safety valve and lever ) 
l = lever for W 1 
e — “ aP >in inches. 

*= “ Q) 

Balance the lever over a sharp edge, and the centre of gravity Q is found; 
measure the distance x from the fulcrum G. 


a P e = W l+Q x 27, 


Wl+Q x 

a e 


28, 


W = 29, 

1 *1 P & Q X QA 

1 W ’ 


Example. Area of the safety valve a = 9 square inches, e = 4£ inches, 
W= 50 pounds, weight of the lever and safety valve <2=15 pounds, and x = IV 
inches. Required at what distances l, l' and l" will the weight W indicate pres¬ 
sures of P = 30, P' — 40, and P" = 50 pounds ? 


Z = 


9X30X4*5-15X17 

50 


= 19*2 inches, 


from the fulcrum Cthe weight Ifwill indicate P — 30 pounds 
V = 37 - 9 inches, when P' = 40 pounds. 

I" = 45-8 “ “ P' = 50 “ 

and thus the lever can be graduated. 















Expansion op Cast Ikon . 


409 


Linear Expansion or Contraction in Indies 
of Cast Iron, Lengths in Feet. 


Difference in Temperature.—Fahrenheit. 



100 ° 

150 ° 

300 ° 

350 ° 

300 ° 

4 : 00 ° 

500 ° 

600 ° 

800 ° 

Feet. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

1 

0.0072 

0.0110 

0.0150 

0.0192 

0.0237 

0.0336 

0.0444 

0.0561 

0.0787 

2 

0.0144 

0.0220 

0.0300 

0.0384 

0.0474 

0.0032 

0.0885 

0.1123 

0.1574 

3 

0.0216 

0.0330 

0.0450 

0.0576 

0.0711 

0.1008 

0.1332 

0.1684 

0.2361 

4 

0.0288 

0.0440 

0.0600 

0.0768 

0.0948 

0.1344 

0.1776 

0.2246 

0,3148 

5 

0.0360 

0.0550 

0.0750 

0.0960 

0.1185 

0.1680 

0.2220 

0 2805 

0.3935 

6 

0.0432 

0.0660 

0.0900 

0.1152 

0.1422 

0.2016 

0.2664 

0.3368 

0.4722 

7 

0.0504 

0.0770 

0.1050 

0.1344 

0.1659 

0.2552 

03108 

0.3929 

0.3509 

8 

0.0576 

0.0880 

0.1200 

0.1536 

0.1896 

0.2688 

0.3552 

0.4496 

0,6396 

9 

0.0648 

0.0990 

0.1350 

0.1728 

0.2133 

0.3024 

0.3996 

0.5052 

0.7083 

10 

0.0720 

0.1102 

0.1502 

0.1926 

0.2376 

0.3360 

0.4440 

0.5616 

0.7872 

11 

0.0792 

0.1214 

0.1652 

0.2125 

0.2615 

0.3696 

0.4884 

0.6177 

0.8659 

12 

0.0864 

0.1316 

0.1802 

0.2318 

0.2853 

0.4032 

0.5328 

0.6739 

09446 

13 

0.0936 

0.1417 

0.1952 

0.2510 

0.3090 

0.4368 

0.5772 

0.7300 

1.0233 

14 

0.1008 

0.1519 

0.2102 

0.2703 

0.3328 

0.4704 

0.6216 

0.7862 

1.1020 

15 

0.1080 

0.1620 

0.2253 

0.2895 

0.3565 

0.5040 

0.6660 

0.8423 

1.1808 

16 

0.1152 

0.1722 

0.2403 

0.30^8 

0.3803 

0.5376 

0.7104 

0.8985 

1.2595 

17 

0.1224 

0.1823 

0.2553 

0.3280 

0.4040 

0.5712 

0.7548 

0.9516 

1.3382 

18 

0.1296 

0.1925 

0.2703 

0.3472 

0.4278 

0.6048 

0.7992 

1.0108 

1.4169 

19 

0.1368 

0.2026 

0.2853 

03665 

0.4515 

0.6384 

0.8436 

1.0669 

1.4956 

20 

0.1440 

0.2203 

0.3005 

0.3852 

0.4752 

0.6720 

0.8880 

1.1232 

1.5744 

21 

0.1512 

0.2305 

0.3155 

0.4045 

0.4995 

0.7056 

0.9324 

1.1793 

1.6531 

22 

0.1584 

0.2407 

0.3275 

0.4238 

0.5228 

0.7392 

0.9768 

1.2394 

1.7318 

23 

0.1656 

0.2508 

0.3425 

0.3430 

0.5465 

0.7728 

1.0212 

1.2915 

1.8105 

24 

0.1728 

0.2610 

0 3575 

0.3623 

0.5703 

0.8064 

1.0656 

1.3477 

1.8892 

25 

0.1800 

0.2711 

0.3725 

0.3815 

0.5940 

0.8400 

1.1100 

1.4038 

1.9679 

26 

0.1872 

0.2813 

0.3876 

0.4008 

0.6179 

0.8736 

1.1544 

1.4600 

2.0467 

27 

0.1944 

0.2914 

0.4026 

0.4200 

0.6415 

0.9072 

1.1988 

1.5161 

21254 

28 

0.2016 

0.3016 

0.4176 

0.4393 

0.6553 

0.9408 

1.2432 

1.5723 

2.2041 

29 

0.2088 

0.3117 

0.4326 

0.4585 

0.6890 

0.9744 

1.2876 

1.6284 

22^29 

30 

0.2160 

0.3304 

0.4507 

0.5778 

0.7128 

1.0080 

1.3320 

1.6848 

2.3616 

31 

0.2232 

0.3405 

0.4657 

0.5970 

0.7365 

1.0416 

1.3764 

1.7409 

2.4403 

32 

0.2304 

0.3507 

0.4807 

0.6163 

0.7603 

1.0752 

1.4208 

1.7971 

2.5190 

33 

0.2376 

0.3608 

0.4957 

0.6355 

0.7841 

1.1088 

1.4652 

1.8533 

2.5977 

34 

0.2448 

0.3710 

0.5107 

0.6548 

0.8078 

1.1424 

1.5096 

1.9094 

2.6764 

35 

0.2520 

0.3811 

0.5258 

0.6740 

0.8316 

1.1760 

1.5540 

1.9656 

2.7552 

36 

0.2592 

0.3913 

0.5408 

0.6933 

0.8553 

1.2096 

1.5984 

2.0217 

2.8339 

37 

0.2664 

0.4014 

0.5558 

0.7125 

0.8791 

1.2432 

1.6428 

2.0779 

2.9126 

38 

0.2736 

0.4116 

0.5708 

0.7298 

0.9028 

1.2768 

1.6872 

2.1340 

2.9913 

39 

0.2808 

0.4217 

0.5858 

0.7490 

0.9266 

1.3104 

1.7316 

2.1902 

3.0701 

40 

0.2880 

0.4406 

0.6009 

0.7704 

0.9504 

1.3440 

1.7760 

2.2464 

3.1488 

45 

03240 

0.4957 

0.6760 

0.8667 

1.0692 

1.5120 

1.9980 

2.5272 

3.5424 

50 

0.3600 

0.5508 

0.7512 

0.9630 

1.1880 

1.6800 

2.2200 

2.8080 1 

3.9360 

55 

0.3960 

0.6059 

0.8263 

1.0593 

1.3068 

1.8480 

2.4420 

3.0888 

4.3296 

60 

0.4230 

0.6610 

0.9014 

1.1556 

1.4256 

2.0160 

2.6640 

3.3696 

4.7132 

65 

0.4680 

0.6665 

0.9765 

1.2519 

1.5444 

2.1840 

2.8860 

3.6540 , 

5.1068 

70 

0.5040 

0.7711 

1.0517 

1.3482 

1.6632 

2.3520 

3.1080 

3.9312 

5.5104 

75 

0.5400 

0.8262 

1.1268 

1.4445 

1.7820 

2.5200 

3.3300 

4.2120 

5.9010 

80 

0.5760 

0.8813 

1.2019 

1.5408 

1.9008 

2.6S80 

3.5520 

4.4948 

6.2976 

85 

0.6120 

0.9364 

1.2770 

1.6371 

2.0196 

2.7560 

3.7740 

4.7756 

6.6912 

90' 

0.6480 

0.9914 

1.3521 

1.7334 

2.1384 

3.0240 

3.9960 

5.0544 

7.0848 

95 

0.6840 

1.0465 

1.4272 

1.8297 

2.2572 

3.1920 

4.2180 

5.3352 

7.4784 

100 

0.7200 

1.1016 

1.5024 

1.9260 

2.3760 

3.3600 

4.4400 

5.6160 

7.8720 

0.00000600 

612 

626 

642 

660 

700 

*740 

780 

820 


Expansion per Degree,Fahrenheit. 


Multiply by 1.1 for wrought iron, 1.5 for copper, 1.6 for brass and 2.6 for zinc. 














































410 


Horse-Power. 


Table of Pressure and Temperature of Steam, 

calculated by the Alexander formula, with a slight modification to accommodate the 
volumes of the component gases, oxygen and hydrogen. 


Pressure 
per sq, in. 

Temp. 

Fahr. 

p. 

T. 

100 

328.2 

150 

358.6 

200 

381.5 

250 

400.0 

300 

415.6 

350 

429.2 

400 

441.1 

4i0 

452.2 

500 

462.1 

600 

479.6 


Pressure 
per sq. in. 


p. 

700 

800 

900 

1000 

1500 

2000 

2500 

3000 

3500 

4000 


Temp. 

Fahr. 


T. 

503.0 

508.4 
520.6 

531.8 

576.4 
610.0 

637.2 

651.3 
680.1 

697.8 


Pressure 
per sq. in. 


P. 

4500 

5000 

6000 

7000 

8000 

9000 

10,000 

15,000 

20.000 

25,000 


Temp. 

Fahr. 


T. 

714.1 

728.3 
754.0 

776.5 

796.4 

814.3 

830.6 

896.2 

945.4 

987.7 


Pressure 
per sq. in. 


P. 

30,000 

35,000 

40,000 

45,000 

50,000 

60,000 

70,000 

80,000 

90,000 

100,000 


Temp. 

Fahr. 


T. 

1019 

1048 

1074 

1098 

1119 

1157 

1190 

1219 

1245 

1285 


In the above table it is supposed that the steam is superheated from an indefi¬ 
nite volume of water. The formulas on pages 394 and 395 are not reliable above 
a pressure of 500 pounds to the square inch. 

The formula of Messrs. Fairbairn and Tate, for the volume of steam, is incon¬ 
sistent with the physical laws involved. The steam in Fairbairn and Tate’s exper¬ 
iments has evidently been moist with globes of water, which made the steam- 
volume too small and the formula wrong. 

For low pressure and temperatures of aqueous vapor, see Ilygrometry, page 357. 

HORSE-POWER IN STEAM-ENGINES. 

Horse-power in machinery is assumed to be about the effect a horse is able to 
produce, and has been estimated and established by Mr. Watt to be 33,000 lbs. 
raised one foot per minute for one horse, which will be the same as 550 lbs. raised 
one foot per second. Mr. Watt adopted a standard steam-pressure of 7 lbs. per 
square inch, established a simple rule for the nominal horse-power of engines, 
which is, “ The square of the diameter of the cylinder in inches multiplied by the 
cube root of the stroke in feet, and divided by the constant number 47, is the nominal 
horse-power. This rule agreed very near to the actual performance of engines in 
those days, but as the improvements advanced we found that the steam-piston can 
move with a greater velocity, and the steam-pressure gradually increased—that 
our day’s engines greatly exceed the above rule. 

Nominal Horse-Power. 

Assume a standard steam-pressure of 30 lbs. per square inch expanded two- 
thirds, the velocity of the steam-piston to be 200j^'$~feet and revolutions per 

minute n = --- - - ’ we will arrive at a formula of nominal horse-power. 

D 2 fS 


li¬ 


lt) 


, for condensing engines, which will agree very near with the actual 

The following tables are calcu- 


performance of our present condensing engines, 
lated from this formula. 

For high-pressure engines I will assume the steam-prossure to be 80 lbs. persquare 
inch, expanded one-liaif, which will give the nominal horse-power— 

. 

a --, high-pressure engines. 

4 

The horse-power iji the accompanying table, divided by 0.4, gives the nominal 
power of high-pressure engines. The diameters D are contained in the first col¬ 
umn in inches, and the stroke S in feet and inches on the top line. 

Indicated. Horse-Power 

Is that imparted by the steam on the cylinder-piston, without friction and workine 
the pumps. 
































Horse Power. 


HI 


ACTUAL HORSE POWER. 

One actual horse power is 33000 lbs. raised one foot in one minute. This 
applied to steam engines will be the mean steam pressure on cylinder 
piston in pounds, multiplied by the velocity of piston in feet per minute, 
divided by 33,000, is the horse power imparted by the steam. From this 
we shall deduct 25 per cent, in condensing engines, and 13T per cent, 
in high pressure engines, for working friction and pumps, the balance 
to be termed the actual horse power. 

Example 1. Fig. and formuke318. Area of steam cylinder A =1809 square 
inches, stroke of piston S=4 feet, indicated pressure of steam 30 lbs. to 
which add the atmospheric pressure 15 lbs. or P — 45 lbs. expanded §, the 
mean pressure will be F=31-459 lbs. (see Expansion Table I.), vacuum 
« = 12 lbs. the engine making tc= 45 revolutions or double stroke per 
minute. Required the actual horse power, H=1 W=- 31-459 y 12-14-7= 

28-759 lbs. 1809V4X28-759Y45 

H= --^ ^ = 425-6 horses. 

22000 

In this example the actual horse power is 11-6 per cent, more than the 
nominal power from the table. 

Example 2. Fig. 318. A high pressure engine of cylinder piston A=314 
square inches, stroke S =3 feet, steam pressure 80 lbs, per square inch, to 
which add 15 lbs. P=95 lbs. expanded i, the engine making n= 56 revo> 
lutions per minute. Required the actual horse power! From the ex¬ 
pansion table we have the mean pressure F = 80-412 lbs., from which sub¬ 
tract the atmospheric pressure 14-7 lbs. W= 65-712 lbs. 


H- 


314X3X65-12X56 

19000 


180-8 horses. 


Annular Expansion Double Cylinder, Fig. 319. 

These kind of engines are now sometimes made in Europe with a view 
to economise fuel, and to extend the expansion of steam. The outer 
cylinder A, A, is annular, similar to that made by Mouslay, but in this 
case it is employed only for expansion, the inner cylinder a is used for high 
pressure only. It is so arranged by steam valves that the high steam is 
acting the whole stroke on the small piston a, after which it is conducted 
to the annular cylinder where it acts expansively on the large piston A, A. 
The two pistons being connected by rods to one common crosshead as 
shown by Fig. 319. This arrangement has been successfully carried out 
by Mr. Jiigerfelt in Nykoping, Sweden. The inner cylinder can be con¬ 
sidered an ordinary high pressure engine where the utilized steam is set 
free into the air at each stroke; but in this case the exhaust steam ac¬ 
complishes a second engagement in the annular cylinder, which according 
to the grade of expansion may greatly exceed the original effect im¬ 
parted in the small cylinder during the first engagement. 

] Example 3. Fig. 319. Area of the high pressure cylinder piston 
a=254-4 square inches, the annular expansive piston A =763-2 square 
inches, stroke of pistons S=3 feet, the high steam pressure P~ 60 lbs. 
vacuum v=12 lbs., making «=65 revolutions per minute. Required the 
actual horse power of the engine H=l The grade of expansion will be 
763*2 

1- -= §, for which the mean pressure on the annular piston will be 

254"4 

jf=32-62 lbs. See Expansion Table II. The effective pressure on the two 
pistons will be F=763-2 (32-62+12—14-7) + 254-4 (60—32-62) = 29800 lbs. 

If=29800X3X65 =264 hor8eg . 

22000 

Example 4. Now we will reject the annular expansion cylinder, and 

I take the effect of the steam without expansion, when the effectual pres- 

I sure will be 60—14-7=45-3 lbs. and the actual power, 

!. H _ ^-4X3X 4 5 -3X66 =118horseg 

l 


19000 














412 


Horse Power. 


If we jonsider the last result as unit we shall have 264—118=146 horses 
or nearly 124 per cent, gained by the expansion, omiting the loss of steam 
in the steam passages. 

In the first case about 11 per cent, was gained by vacuum, but that ad¬ 
vantage is rather in favour of the utility of expansion, because the high 
steam cannot so well be introduced into the condenser. 

The economy will be in the same proportion when the same grade of 
expansion is used in one cylinder. 

I do not mean to maintain that this high per centage of economy is al¬ 
ways fully realized in practice, as I am well aware of cases where expan¬ 
sion is of little use, caused by misconception and carelessness in its em¬ 
ployment. There are many circumstances about an engine which are in 
favour of expansion, for instance, the steam passages between the main 
valve and cylinder, and the clearance between the piston and cylinder 
heads, contains a great deal of steam which is a total-loss, but when ex¬ 
pansion is used, that steam expands into the cylinder, and is consequently 
utilized. The expanded exhaust require a smaller air pump than would 
he necessary for high steam introduced in the condenser. 


Half Trunk Expansion Engines. Fig. 320. 

This kind of engines has been introduced by Mr. Carlsund, and are ex - 
tensiveiy used in Sweden, they are well suited for Gunboats where the 
machinery is required to be below the water line. The high steam is em¬ 
ployed throughout the stroke in the annular space around the trunk, 
after which it is conducted to act expansively on the large piston A 
Fig. 320. 

Example 5. Fig. 320. Area of the annular piston a=662 square inches, 
and A=2248 square inches, stroke of piston S=4 feet, steam pressure 
P =90 lbs., making n= 6b revolutions per minute. Required the actual 
horse power? 

662 

Grade of expansion = 1— — = ¥, 

2248 ’ 

From the Expansion Table II. we have/=41-58 lbs. mean pressure on A. 

The effectual pressure w r ill be F=2248 (41-68—14-7) -j- 562 (90,—41-68) = 
87639 lbs., high pressure 87639Y4Y68 

H — - —- — - = 627-3 horses. 

38000 


Double Cylinder Expansion Engines, Fig. 321. 

This kind of engines are now made in England and are said to be very 
economical. The small cylinder is used for high pressure, from which 
the steam is conveyed to expand in the large cylinder. In the figure it 
is arranged so that the pistons follow one ancrther in one direction, when 
the steam must be conveyed from the top of the small cylinder to the 
bottom of the large one, and vice-versa ; but it is sometimes arranged so 
that the pistons move in opposite direction, when the steam is conveyed 
direct at the same ends from the small cylinder to the large one, which 
has the advantage of making the steam passages shorter, but is more 
complicated in concentrating the motion. 

Example 6. 

High pressure cylinder, { « = ^square inches. 


Expansion cylinder, 


f A = 3848 square inches. 
( S = 10 feet. 


Steam pressure in the small cylinder P=40 lbs., vacuum v=^12 lbs - ., 
making n =21 revolutions per minute. Required the actual horse 
power, if=? 

Grade of expansion =1 — 


962X5 


3848X10 

From the Expansion Table II. we have/=11.879 lbs., mean pressure on A. 
3848X10 (11.879 + 12-14:7)+962X5 (40-11.879) = 366767 lbs. of mo¬ 
mentum. 366767X21 

H = —-= 350 horses. 

22000 















Horse Power of Engines. 


413 



318. One double acting Cylinder. 


Actual 
horse 
power . 


H= 

H= 


A S W n ^ 

22000 
A S Wn } 
19000 .'i 


cond. engs. 

high pr. en¬ 
gines. 


W=F-\-v —14-7 for cond. engines. 
W—F —14-7 forliiglipressure engines. 






319. Annular expansion double Cylinder. 

F—~ a L2'3 (log.4 — log.a)-$-l]. 

A 


F A—P a V=A(f±v—U-1) 
f~ A—a 1 + a (P—f)- 


Actual 
horse •s 
power. 1 

r VSn 

1 a ~ 22000’ cond - 
1 rr VSn T . , 

^ 19000 ’ 1Ughpr ' engS ' 

320. Halftrunk expansion Cylinder. 

F=^ l2-3(log.A—log.a)+l-\. 

A 

Fa-Pa V=A(fA-v —14’7) 

J ~ A 

Actual ' 
horse M 
power. | 

L—a * -1 

, VSn 

f //—- 1 cond. engines. 

) 44000 

VSn , . , 

H— ’ high pr. engs. 

1 38000 fo F 




AS 



321. Double Cylinder expansion. 

F,= ^Fci-ZClog.AS-logMs)+\-] 

-A /b 

FA — Pa w=AS(f-\-v —14*7J 

A—a ’ + a s ( p —f)‘ 

( TT W n ' cond. engines. 
Actual J 22000 
horse < 

power. I H== J^L, higlipr. engine*. 




















































































414 


Nominal Hoksepoweb op Condensing Engines. 


Siam 


Stroke of Cylinder Piston S in feet, 


IJ 

V 

V 3" 1' 6" V 9"\ 2 • 

|2' 3' 

'| 2' G" 2' 9" 3' 

3' 6" 4' 

4' 6"[ S’ 

in. 

; h 

H 

H 

H 

H 

H 

H 

1 H 

H 

H 

If 

1 H 

I M 

6 

3-6 

3*88 

4*12 

4*33 

4'5r 

S 4-75 

4*8! 

3 5*0' 

4 5*1 

9 5*4 

7 5-7 

1 5*94 6*16 

7 

4-9 

5*27 

5*61 

5*90 

6*17 

6*43 

6*6, 

3 6-8( 

1 7*0 

1 7*4* 

4 7*78 8*1 

0! 8*38 

8 

6-4 

6*90 

7*32 

7*71 

8*0 C 

8*39 

8-6{ 

3 8*9( 

5 9*2, 

1 9-7- 

2 10* 

11 10* 

6 11*0 

9 

8-1 

8*72 

9*27 

9-75 

10*5 

10-6 

11*( 

) u*: 

1 11*' 

r 12*; 

1 12* 

9 :13- 

4 13*9 

10 

10 

10-8 

11*4 

12*0 

12*6 

13*1 

13*6 

5 14*( 

14*^ 

1 154 

> 15* 

9 16* 

5 17*1 

11 

12-1 

13*0 

13*9 

14*6 

15*2 

15*8 

16*4 

1 16*9 

17*4 

1 184 

19* 

2 20* 

) 20*7 

12 

14-4 

15*5 

16*5 

17-4 

18*1 

18*9 

19*5 

20*9 

204 

21*9 

22- 

9 23*; 

3 24*6 

13 

16-9 

18*2 

19*3 

20*3 

21*3 

22*1 

22*9 

237 

24*4 

25*1 

26\ 

3 27*9 

) 28*9 

14 

19-6 

211 

22-4 

23*6 

24*7 

25*7 

26 e 

27*4 

284 

29*7 

31*, 

l 32*^ 

1 33*5 

15 

2-25 

24*2 

25*8 

27*1 

28*3 

29*5 

30*5 

31*5 

32*4 

34*1 

35*7j 37*1 

38*5 

16 

256 

27*4 

29*3 

30*8 

32-2 

33*5 

34*7 

35*8 

37*0 

38*9 

40*6 424 

43*8 

17 

28-9 

311 

33*1 

34*8 

36*4 

37-9 

39-2 

40*5 

41*7 

43*9 

45*9 

47*7 

49*4 

18 

32-4 

34*9 

37*1 

39*0 

40*8 

42*5 

44*0 

45*4 

46*8 

49*2 

51-4 

53*£ 

55*4 

19 

36-1 

38-9 

41-3 

43*5 

45-5 

47-3 

49*0 

50*6 

52*1 

54*8 

57*2 

59*<: 

61*7 

20 

40-0 

43*1 

45*8 

48*2 

50*4 

52*4 

54*3 

56*0 

57*7 

60*7 

63*5 

66*0 

68*4 

21 

449 

47*5 

505 

53*1 

55*6 

57-8 

59-8 

61*7 

63*6 

67*0 

70*0 

72-8 

75.4 

22 

48-4 

52*1 

55*4 

58*3 

61-0 

63*4 

65*6 

64*8 

69*8 

73*5 

76*8 

80-0 

82*8 

23 

52-9 

57-0 

60*5 

63*7 

66*7 

69*3 

71*8 

74*1 

76*3 

80*3 

84*0 

87*4 

90*5 

2 4 

57-6 

62*0 

65*9 

69*4 

72*6 

75*5 

78*1 

80-7 

83-1 

87*4 

91*5 

95*2 

98*6 

25 

62*5 

67*3 

71*5 

75*3 

78*7 

81*9 

84-8 

87*5 

90*2 

94-8 

99*2 

103 

107 

26 

67*6 

72*8 

77*3 

81-5 

85*2 

88*6 

91*7 

94*7 

97*5 

102 

107 

111 

116 

27 

72*9 

78-5 

83*5 

87*8 

91*9 

95*6 

99*0 

102 

105 

111 

116 

120 

125 

28 

78*4 

84-4 

89*8 

94*5 

98*8 

102 

106 

110 

113 

119 

124 

129 

134 

2 9 

84J 

90-5 

96-2 

101 

106 

110 

114 

118 

121 

128 

133 

139 

144 

30 

90*0 

96-9 

103 

108 

113 

118 

122 

126 

130 

137 

143 

149 

154 

31 

96*1 

103 

110 

116 

121 

126 

130 

134 

139 

146 

153 

159 

164 

32 

102 

110 

117 

123 

129 

134 

138 

143 

148 

155 

163 

170 

175 

33 

109 

117 

124 

131 

137 

142 

147 

152 

157 

165 

173 

ISO 

186 

34 

115 

124 

132 

139 

145 

151 

157 

162 

167 

175 

183 

190 

198 

35 

122 

132 

140 

148 

154 

160 

166 

172 

177 

186 

194 

202 

210 

36 

129 

140 

148 

156 

163 

170 

176 

182 

187 

197 

205 

214 

222 

37 

137 

147 

156 

165 

172 

180 

186 

192 

198 

208 

217 

226 

234 

38 

144 

155 

165 

174 

182 

190 

196 

202 

209 

218 

229 

238 

247 

39 

152 

164 

174 

183 

192 

200 

206 

213 

220 

231 

241 

251 

260 

40 

160 

172 

183 

193 

202 

210 

217 

224 

231 

243 

254 

264 

274 

42 

176 

190 

202 

212 

222 

231 

240 

347 

254 

268 

280 

291 

302 

44 

193 

208 

221 

233 

244, 

254 

263 

271 

280 

294 

307 

320 

331 

46 

211 

228 

242 

255 

266 

277 

287 

297 

306 

321 

336 

350 

362 

48 

230 

248 

264 

277 

290 

302 

313 

323 

332 

350 

366 

380 

394 

50 

250 

269 

286 

301 

315 

328 

339 

350 

360 

380 

397 

413 

427 

52 

270 

291 

309 

326 

340 

354 

367 

378 

390 

410 

429 

446 

463 

54 

291 

314 

333 

351 

367 

382 

396 

408 

420 

443 

463 

481 

500 

60 

360 

388 

412 

433 

453 

472 

488 

504 

519 

547 

571 

594 

616 

66 

435 

468 

498 

525 

548 

571 

591 

610 

628 

661 

690 

718 

744 

72 

518 

558 

593 

626 

653 

679 

704 

726 

748 

787 

822 

856 

886 

78 

608 

655 

696 

734 

766 

784 

825 

852 

877 

924 

964 

1003 

1039 

84 

705 

759 

807, 

851 

888 

924 

957 

989 

1015 

L0 71 

1116 

1166 

1206 

90 

810 

872 

927 

975 

L020 

1062 

1100 

1134 

1168 

1229 

1284 

1336 

L385 

96 

921 

991 

1053illl0 

L160 

L206 

L249I: 

1291 

1327 

1400 

L460 

1505 

L575 















































































Nominal Horsepower of Condensing Engines, 


415 


Stroke of Cylinder Piston S in feet. 


D 1 

6' 

7' 

S' 

9' 

10 ' 

11 ' 

12 ' 

1 13 ' 

14 ' 

15 ' 

16' 

| IS' 

20 ' 


H 

11 

H 

H 

H 

H 

il 

1 H 

• H 

H 

H 

H 

H 

30 

163 

172 

180 

187 

194 

200 

206 

211 

217 

222 

226 

23C 

244 

3 2 

186 

196 

204 

213 

220 

227 

252 

241 

246 

253 

258 

268 

278 

34 

210 

221 

231 

240 

249 

257 

264 

272 

276 

285 

291 

303 

313 

36 

235 

248 

259 

269 

273 

288 

296 

306 

312 

319 

326 

339 

351 

38 

262 

276 

289 

299 

311 

321 

330 

341 

348 

356 

363 

378 

391 

40 

290 

306 

320 

333 

344 

355 

366 

377 

385 

395 

403 

419 

434 

42 

320 

336 

352 

365 

380 

392 

404 

416 

425 

435 

444 

462 

478 

44 

352 

371 

387 

402 

417 

430 

453 

461 

466 

477 

494 

507 

525 

46 

384 

405 

423 

440 

460 

470 

4S4 

497 

510 

522 

533 

554 

587 

48 

418 

441 

461 

479 

496 

512 

527 

541 

555 

569 

580 

603 

625 

50 

551 

478 

500 

520 

538 

555 

572 

588 

602: 617 

630 

655 

677 

52 

491 

518 

54] 

562 

582 

601 

619 

635 

651 

667 

681 

708 

733 

54 

529 

558 

583 

606 

628 

648 

667 

685 

700 

719 

734 

764 

790 

56 

570 

600 

637 

652 

675 

697 

718 

737 

755 

775 

790 

811 

850 

58 

611 

644 

673 

700 

724 

648 

770 

791 

810 

830 

847 

880 

912 

60 

654 

689 

720 

749 

775 

800 

824 

846 

867 

888 

907 

943 

977 

62 

698 

736 

769 

800 

828 

855 

879 

903 

925 

948 

968 

1007 

1043 

64 

744 

784 

819 

852 

882 

911 

938 

963 

987 

10101024 

1073 

1101 

66 

791 

834 

871 

906 

938 

968 

997 

1024 

1049110741099 

1141 

1182 

68 

840 

885 

925 

960 

996 

1023 

1059 

1087 

11141140 1165 

1211 

1254 

70 

890 

938 

980 

1019 

1055 

1089 

1122 

1152 

1171 

1208 1234 

1283 

1329 

72 

942 

994 

1037 

1078 

1116 

1153 

1187 

1218 

1249 

1278 1306 

1358 

1406 

74 

995 

1048 

1095 

1139 

1179 

1218 

1254 

1287 

1319 

1350 

1380 

1434 

1485 

76 

1050 

1105 

1155 

1201 

1244 

1284 

1322 

1358 

1392 

1424 

1455 

1512 

1567 

78 

1105 

1165 

1219 

1265 

1310 

1353 

1393 

1430 

1466 

1500 

1533 

1594 

1649 

80 

1162 

1225 

1280 

1331 

1378 

1423 

1465 

1504 

1542 

1578 

1612 

1676 

1737 

84 

1282 

1350 

1411 

1467 

1520 

1569 

1615 

1658 

1700 

1741 

1778 

1848 

1914 

88 

1407 

1423 

1549 

1610 

1668 

1722 

1773 

1820 

1866 

1909 

1951 

2029 

2100 

92 

1538 

1619 

1693 

1761 

1823 

1882 

1938 

1990 

2039 

2086 

2133 

2258 

2297 

96 

1674 

1763 

1843 

1917 

1985 

2049 

2010 

2166 

2221 

2272 

2322 

2414 

2474 

100 

1817 

1913 

2000 

2080 

2154 

2224 

2290 

2351 

2400 

2466 

2520 

2620 

2714 

104 

1964 

1969 

2163 

2250 

2349 

2405 

2477 

2542 

2608 

2666 

2725 

2833 

2935 

108 

2119 

2231 

2333 

2426 

2512 

2594 

2671 

2742 

2806 

2873 

2939 

3056 

3165 

112 

2279 

2399 

2509 

2609 

2702 

2790 

2871 

2949 

3023 

3092 

3161 

3286 

3404 

116 

2445 

2574 

2691 

2799 

2898 

2992 

3081 

3163 

3243 

3315 

3391 

3525 

3651 

120 

2616 

2754 

2880 

2995 

3312 

3202 

3297 

3385 

3471 

3550 

3628 

3772 

3908 

124 

2793 

2941 

3075 

3198 

3101 

3419 

3521 

3614 

3706 

3790 

3874 

4028 

4172 

128 

'2977 

3133 

3277 

3408 

3529 

3643 

3752 

3852 

3949 

4038 

4128 

4292 

4446 

132 

3166 

3333 

3485 

3624 

3753 

3875 

3990 

4096 

4190 

4295 

4390 

4565 

4728 

136 

3360 

3538 

3699 

3847 

3984 

4113 

4235 

4348 

4457 

4557 

4654 

4846 

5019 

140 

3561 

3749 

3920 

4077 

4222 

4359 

448S 

4608 

4716 

4832 

4939 

5135 

5319 

144 

3767 

3966 

4147 

4313 

4466 

4611 

4748 

4875 

4997 

5111 

5225 

5432 

5681 

148 

3980 

4190 

4381 

4556 

4718 

4871 

5016 

5179 

5279 

5399 

5519 

5739 

5944 

152 

4198 

4402 

4621 

4805 

4976 

5138 

5291 

5431 

5568 

5696 

5821 

6053 

6270 

156 

4421 

4655 

4867 

5062 

5242 

5412 

5573 

5721 

5865 

6000 

6132 

6376 

6604 

162 

4768 

5020 

5249 

5458 

5653 

5836 

5010 

6170 

6324 

6469 

6613 

6876 

7122 

168 

5128 

5399 

5645 

5870 

6079 

6277 

6463 

6635 

6802; 6958 

7112 

7094 

1659 

174 

5494 

5791 

6055 

6297 

6521 

6733 

6933 

71171 

7296 7464 

7629 

7932 8216 

180 J 

5887] 

6198)6480 

6539 6979 

7205) 

7419 

7617|7809 798-9 

8164) 

S488.8793 




























































41G 


Approximate Horse-Fower. 


Approximate Horse-Power 

of small high-pressure engines. H — 0.1 D- \/ S. Steam pressure not less than SO 

pounds to the square inch. 


Diam. 

D 

Inches. 

3 

4r 

5 

6 

Strok* 

7 

3 S of 

8 

pisto 

9 

n in ii 

10 

iches. 

13 

14: 

15 

16 

18 

2 

.577 

.634 

.684 

.727 

.765 

.800 

.832 

.862 

.915 

.964 

.985 

1.00 

1.05 

2£ 

.900 

.990 

1.07 

1.13 

1.20 

1.25 

1.30 

1.34 

1.42 

1.50 

1.54 

1.57 

1.63 

3 

1.30 

1.43 

1.54 

1.64 

1.72 

1.80 

1.87 

1.94 

2.06 

2.17 

2.22 

2.27 

2.36 

3j 

1.77 

1.94 

2.10 

2.22 

2.34 

2.45 

2.55 

2.64 

2.80 

2.95 

3.00 

3.69 

3.2! 

4 

2.31 

2.54 

2.74 

2.90 

3.06 

3.20 

3.33 

3.44 

3.66 

3.85 

3.94 

405 

4.19 

44 

2.92 

3.21 

3.47 

3.68 

3.87 

4.05 

4.12 

4.36 

4.64 

4.88 

5.00 

5.10 

5.JO 

5 

3.60 

3.96 

4.27 

4.54 

4.78 

5.00 

5.20 

5.38 

5.72 

6.02 

6.16 

6.30 

6.55 

6 

5.19 

5.70 

6.15 

6.53 

6.89 

7.20 

7.55 

7.82 

8.31 

8.75 

8.95 

9.15 

9.50 

7 

7.08 

7.78 

8.40 

S.92 

9.40 

9.80 

10.2 

10.6 

11.2 

11.8 

12.1 

12.3 

12.9 

8 

9.25 

10.1 

11.0 

11.6 

12.2 

12.8 

13.3 

13.8 

11.6 

15.4 

15.7 

16.1 

16.8 

9 

11.7 

12.9 

13.9 

14.7 

15.5 

16.2 

16.8 

17.4 

1S.5 

19.5 

20.0 

20.4 

21.2 

10 

14.4 

15.9 

17.1 

18.2 

19.1 

20.0 

20.8 

21.5 

22.9 

24.1 

21.6 

25.2 

26.2 

11 

17.5 

19.2 

20.8 

22.0 

23.2 

24.2 

25.2 

26.1 

27.7 

29.2 

29.9 

30.5 

31.6 

12 

20.8 

22.9 

24.7 

26.2 

27.6 

2 S.8 

30.0 

31.0 

33.0 

34.8 

«jO.O 

36.3 

37.8 


The horse-power of small engines, as counted by the English, is only 0.4 of 
that in this table for the same size cylinders. 

To Approiimate tlie Size of Steam-Engines. 

Example 1. It is required to build a river steamer of displacement T = 1000 
tons to run M = 16 nautical miles per hour. Required, the size of the cylinder 
for an ordinary overbeam engine? From the table of steamship performance will 
be found the required actual power H = 1798 horses. 

From the table of Nominal horse-power select the approximate size of cylinder, 
which may be D — 88 inches, diameter of cylinder by S — 14 feet stroke, which 
answers to II — 1866 horses nominal. In this case the nominal horse-power can 
be considered the same as the actual. 

Example 2. A propeller steamer is to run M = 10 nautical miles per hour, with 
a displacement T = 3400 tons. Required, the size of .the cylinders? 

From table of steamship performance H == 992 horses, to be divided into two 
cylinders of 406 each. Select from table of Nominal horse-power D = 60 inches 
diameter of cylinders and N = 2' 19" stroke of piston, which answers to H = 504, 
or 504 X 2 = 1008 horses of the two cylinders. After these approximations are 
made, make a careful calculation from the original formulas. 

Example 3. Suppose the propeller for the steamer in the preceding Example 2 
makes n = 60 revolutions per minute. Required, the diameter of the propeller- 
shaft? See Table, page 418, for wrouglit-iron shafts, for 1000 horses and 60 revolu¬ 
tions, the shaft should be 12.8 inches. 

Example 4. A steamer of 7=2500 tons is to run J/ = 9 nautical miles per 
hour with an indicated steam-pressure of 20 lbs., or P = 35 lbs. per square inch, 
expanded Required, the consumption of fuel in tons per 24 hours? 

Table of steamship performance H = 585 horses. 

Table Y., page 409, consumption of fuel, 3.44 tons. 

The required consumption will be 5.85 X 3.44 = 20.124 tons per 24 hours’ 
steaming. 










































SotTNTX 


417 


SOUND. 

Velocity of Sound through Air. 

v = velocity in feet per second. 
t — temperature of the air, Fahr. scale. 

D — distance in feet the sound travels in the time T. 


v= 1089.42/l + 0.00208(t — 32). 

Velocity of sound in water is about 4 times that in air, and 8 times that through 
solids. 

Intensity of sound is inversely as the square of the distance. 

D = 1089.42 Ty'l + 0.002080 — 32), 



Example. A ship at sea was seen to fire a cannon, and 6.5 seconds afterward the 
report was heard; the temperature in the air was 60°. Required, the distance to 
the ship. 


D= 1089.42 X 6.5/1 + .00208(60°— 32) =7300 feet, or 1.38 miles. 






Attdible at a distance of 

Descriptions of Sound. 


Feet. 


Miles. 

A powerful human voice in the open air, no wind, 

460 


0.087 

Report of a musket, 

• • 

• • • 

• 

16,000 


3.02 

Drum, 




10,500 


2 

Music, strong brass band, . 

• • • 


15,840 


o 

0 

Cannonading, very strong, 

• • 

• . 

575,000 


90 

In a barely observable breeze a 

strong human voice 





with the wind can be heard. 

. 

• 

15,840 


3 

Ringing Bells—Weight, Dimension and Key-note. 

Place, and when 

W eight. 

Diameter. 

Sound-bow. 

Key- 

Vibrat. 

cast. 

W. 

D. 

S. 

Jc. 

note. 

n. 


pounds. 

inches. 

inches. 

S:D. 

note. 

vib. 

Moscow, 1786, .... 

432,300 

272 

23 

0.084 

D 


18.21 

“ St. Ivan’s. 1817, 

127,350 

185 

14.750 

0.080 

Git 

25.1 

Nishni-Novgorot 1, Russia, 

69,664 

151.25 

12.125 

0.080 

B 


30.58 

Olmutz, Bohemia, . . 

40,320 

121 

9.125 

0.075 


39.4 

Vienna, Austria, . . . 

40,200 

118 

9.500 

0.080 

i»it 

38.6 

Westminster, Rug., 1856. 

35,625 

113.5 

9.375 

0.083 

E 


40.3 

Erfurt, 1487, .... 

30,800 

103.5 

7.75 

0.075 

F 


42 

Paris. 1680, .... 

22,672 

103 

7.5 

0.073 

E 


40 

Montreal. 1847, . . . 

28,560 

103 

7.8 

0.0S2 

F 


42 

York, Eng.. 1845. . . 

24,080 

100 

8 

0.080 

n 


4e 4 

New York, City Hall, 

22,300 

108 

8.3 

0.077 

E 


41 

St. Peter’s, Rome, . . 

18.600 

97.25 

7.5 

0.077 

n 

46 

Oxford, Great Tom, 1680, 

17.024 

S4 

6125 

0.073 

Gt 

50 

Cologne, 1447, . . . 

16,016 

95 

7.2 

0.076 

G 


49 

Brussels, Belgium, . . 

15.848 

81 

5.75 

0.071 

Git 

51 

Lincoln, Eng., 1834, 

12,096 

82.5 

6 

0 073 

G£ 

51 

St. Paul’s, Eng., 1716, . 

11.500 

81 

6.08 

0.075 

A 


54 

Exeter, 1675, .... 

10,080 

76 

5 

0.066 

Git 

50 

Old Lincoln. 1610, . . . 

9,8oG 

75.5 

5.93 

0.078 

i > 


60.4 

Westminster, 1857, . . 

8,960 

72 

5.75 

0.079 

B 


60.4 


























418 


Diameters of Wrougiit-Iron Shafts. 


Diameters in Indies of Wronglit-Iron Shafts. 


Morse Number of revolutions per minute of wrought-iron shafts . 


power 

10 

15 

30 

35 

30 

1 35 

40 

45 

50 

55 

60 

70 

80 

u . 

111. 

Iu. 

Iu. 

In. 

In. 

Iu. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

L 

2.32 

2.07 

1.84 

1.71 

1.61 

1.53 

1.46 

1.41 

1.36 

1.31 

1.28 

1.22 

1.16 

2 

2.92 

2.55 

2.32 

2.16 

2.03 

1.92 

1.84 

1.72 

1.71 

1.66 

1.61 

1.53 

1.46 

3 

3.40 

2.72 

2.66 

2.47 

2.33 

2.21 

2.11 

2.03 

1.91 

1.90 

1.71 

1.75 

1.67 

4 

3.68 

3.22 

2.92 

2.72 

2.55 

2.43 

2.32 

2.23 

2.16 

2.09 

1.80 

. 1.93 

1.84 

5 

3.97 

3.48 

3.15 

2.92 

2.75 

2.61 

2.50 

2.41 

2.33 

2.25 

1.88 

2.08 

1.99 

6 

4.22 

3.68 

3.35 

3.11 

2.92 

2.78 

2.66 

2.56 

2.47 

2.39 

2.06 

2.21 

2.11 

7 

4.44 

3.88 

3.52 

3.27 

3.06 

2.92 

2.80 

2.69 

2.60 

2.52 

2.12 

2.33 

2.22 

8 

4.64 

4.05 

3.69 

3.42 

3.22 

3.06 

2.92 

2.82 

2.72 

2.63 

2.27 

2.43 

2.33 

9 

4.82 

4.22 

3.83 

3.56 

3.35 

3.18 

3.04 

2.93 

2.82 

2.74 

2.40 

2.52 

2.42 

10 

5.00 

4.36 

3.98 

3.69 

3.47 

3.29 

3.15 

3.03 

2.92 

2.83 

2.75 

1.61 

2.50 

12 

5.34 

4.64 

4.22 

3.92 

3.68 

3.50 

3 . 35 - 

3.22 

3.11 

3.01 

2.92 

2.78 

2.61 

15 

5.72 

5.00 

4.54 

4.22 

3.97 

3.76 

3.60 

3.47 

3.35 

3.24 

3.15 

3.00 

2.86 

20 

6.30 

5.50 

5.00 

4.64 

4.37 

4.14 

3.97 

3.82 

3.96 

3.57 

3.47 

3.29 

3.15 

25 

6.79 

5.93 

5.39 

5.00 

4.71 

4.46 

4.28 

4.11 

3.97 

3.84 

3.74 

3.55 

3.40 

30 

7.21 

6.30 

5.72 

5.32 

5.00 

5.74 

4.54 

4.37 

4.22 

4.08 

3.97 

3.77 

3.61 

35 

7.59 

6.63 

6.03 

5.60 

5.26 

5.00 

4.78 

4.60 

4.44 

4.30 

4.18 

3.97 

3.80 

40 

7.94 

6.93 

6.30 

5.85 

5.50 

5.22 

5.00 

4.81 

4.65 

4.50 

4.37 

4.15 

3.98 

45 

8.25 

7.20 

6.55 

6.09 

5.73 

5.43 

5.20 

5.00 

4.83 

4.67 

4.54 

4.31 

4.13 

50 

8.55 

7.47 

6.79 

6.30 

5.93 

5.62 

5.38 

5.18 

5.00 

4.84 

4.71 

4.47 

4.28 

60 

9.08 

7.93 

7.21 

6.70 

6.30 

5.98 

5.72 

5.50 

5.32 

5.15 

5.08 

4.75 

4.55 

70 

9.57 

8.36 

7.59 

7.05 

6.64 

6.30 

6.03 

5.80 

5.60 

5.42 

5.27 

5.00 

4.79 

80 

10.0 

8.75 

7.94 

7.37 

6.94 

6.58 

6.30 

6.07 

5.85 

5.67 

5.51 

5.23 

5.00 

90 

10.4 

9.11 

8.25 

7.67 

7.22 

6.84 

6.55 

6.30 

6.10 

5.90 

5.73 

5.44 

5.20 

100 

10.8 

9.41 

8.55 

7.94 

7.47 

7.09 

6.78 

6.53 

6.31 

6.10 

5.93 

5.63 

5.39 

120 

11.4 

10.0 

9.09 

8.44 

7.95 

7.53 

7.21 

6.94 

6.70 

6.49 

6.31 

6.00 

5.73 

140 

12.0 

11.5 

9.57 

8.89 

8.37 

7.93 

7.59 

7.31 

7.06 

6.83 

6.64 

6.30 

6.03 

160 

12.6 

11.0 

10.0 

9.29 

8.74 

8.29 

7.94 

7.64 

7.38 

7.14 

6.94 

6.59 

6.31 

180 

13.1 

11.4 

10.4 

9.67 

9.09 

8.62 

8.26 

7.95 

7.67 

7.42 

7.21 

6.85 

6.55 

20 0 

13.6 

11.9 

10.8 

10.0 

9.42 

8.94 

8.55 

8.23 

7.94 

7.69 

7.48 

7.10 

6.79 

2:0 

14.6 

12.8 

11.6 

10.8 

10.1 

9.63 

9.22 

8.87 

8.56 

8.29 

8.06 

7.65 

7.32 

300 

15.5 

13.6 

12.3 

11.5 

10.8 

10.2 

9.80 

9.43 

9.10 

8 .SO 

8.57 

8.12 

7.77 

350 

16.3 

14.3 

13.0 

12.0 

11.3 

10.8 

10.3 

9.92 

9.58 

9.27 

9.00 

8.55 

8.18 

400 

17.1 

15.0 

13.6 

12.6 

11.9 

11.2 

10.8 

10.4 

10.0 

9.69 

9.42 

8.94 

8.55 

450 

17.8 

15.5 

14.1 

13.1 

12.3 

11.7 

11.2 

10.8 

10.4 

10.1 

9.80 

9.30 

8.89 

500 

18.4 

16.1 

14.6 

13.6 

12.8 

12.1 

11.6 

11.2 

10.8 

10.4 

10.1 

9.64 

9.21 

550 

19.0 

16.6 

15.1 

14.0 

13.2 

12.5 

12.0 

11.5 

11.1 

10.8 

10.5 

9.94 

9.50 

600 

19.6 

17.1 

15.5 

14.4 

13.5 

12.9 

J 2.3 

11.9 

11.5 

11.1 

10.8 

10.2 

9.79 

700 

20.7 

18.0 

16.4 

15.2 

14.3 

13.6 

13.0 

12.5 

12.1 

11.7 

11.4 

10.8 

10.3 

SOO 

21.5 

18.9 

17.1 

15.9 

15.0 

14.2 

13.6 

13.1 

12.6 

12.2 

11.9 

11.9 

10.8 

1000 

23.3 

20.4 

18.5 

17.1 

16.1 

15.3 

14.6 

14.1 

13.6 

13.2 

12.8 

12.2 

11.6 

1200 

24.7 

21.6 

19.6 

19.2 

17.1 

16.3 

15.6 

14.9 

14.5 

14.0 

13.6 

12.9 

12.4 

1500 

26.6 

23.3 

21.1 

19.6 

18.5 

17.5 

16.7 

16.1 

15.5 

15.1 

14.6 

13.9 

13.3 

2000 

29.3 

25.5 

23.5 

21.5 

20.3 

19.3 

18.4 

17.7 

17.1 

16.6 

16.1 

15.3 

14.6 

2500 

31.5 

27.5 

25.0 

23.3 

21.9 

20.8 

19.8 

19.1 

18.4 

17.9 

17.3 

16.5 

15.8 

3000 

33.5 

29.3 

26.6 

24.8 

23.3 

22.1 

21.1 

20.3 

19.6 

19.0 

18.4 

17.5 

16.7 

3500 

35.2 

30.8 

28.0 

26.0 

24.4 

23.3 

22.2 

21.4 

20.7 

20.0 

19.4 

18.4 

17.6 

4000 

36.8 

32.2 

29.3 

27.2 

25.6 

24.3 

23.3 

22.3 

21.6 

20.9 

20.3 

19.3 

18.5 

4500 

38.4 

33.5 

30.4 

28.3 

26.6 

25.2 

24.1 

23.3 

22.4 

21.7 

21.1 

20.0 

19.2 

*>000 

39.6 

34.7 

31.5 

29.3 

27.2 

26.1 

25.0 

24.1 

23.3 

22.5 

21.9 

20.8 

19.9 


-J 











































Diameters of Wrougiit-Iron Shafts. 


419 


Diameters in Inches of Wrought-Iron Shafts. 


Horse 


Number of revolutions per minute of wrought-iron shafts. 


power . 

100 

125 

150 

175 

200 

250 

300 

350 

400 

500 

600 

800 

1000 

H . 

ha . 

III. 

In. 

la. 

la. 

la . 

In. 

In . 

In. 

In. 

la. 

la. 

In. 

1 

1.08 

1.00 

0.94 

0.90 

0.85 

0.80 

0.75 

0.71 

0.68 

0.63 

0.59 

0.54 

0.5 

2 

1.35 

1.26 

1.19 

1.13 

1.08 

1.00 

0.94 

0.90 

0.86 

0.80 

0.75 

0.68 

0.63 

3 

1.55 

1.44 

1.36 

1.29 

1.24 

1.15 

1.08 

1.03 

0.98 

0.91 

0.86 

0.78 

0.72 

4 

1.71 

1.59 

• 1.50 

1.42 

1.36 

1.26 

1.19 

1.13 

1.08 

1.00 

0.94 

0.86 

0.80 

5 

1.84 

1.71 

1.61 

1.53 

1.46 

1.36 

1.28 

1.22 

1.16 

1.08 

1.02 

0.92 

0.86 

6 

1.96 

1.82 

1.71 

1.62 

1.56 

1.44 

1.36 

1.29 

1.24 

1.15 

1.08 

0.98 

0.91 

7 

2.06 

1.92 

1.80 

1.71 

1.64 

1.52 

1.43 

1.36 

1.30 

1.21 

1.14 

1.03 

0.96 

8 

2.15 

2.00 

1.88 

1.79 

1.71 

1.59 

1.49 

1.42 

1.36 

1.26 

1.19 

1.08 

1.00 

9 

2.24 

2.08 

1.96 

1.86 

1.78 

1.65 

1.55 

1.48 

1.41 

1.31 

1.24 

1.12 

1.04 

10 

2.22 

2.16 

2.03 

1.93 

1.85 

1.71 

1.61 

1.53 

1.47 

1.36 

1.28 

1.16 

1.08 

12 

2.47 

2.29 

2.15 

2.05 

1.96 

1.82 

1.71 

1.63 

1.56 

1.44 

1.36 

1.24 

1.15 

15 

2.66 

2.47 

2.32 

2.20 

2.11 

1.96 

1.85 

1.75 

1.67 

1.55 

1.46 

1.33 

1.24 

20 

2.92 

2.71 

2.56 

2.43 

2.33 

2.15 

2.03 

1.93 

1.84 

1.71 

1.61 

1.46 

1.36 

25 

3.15 

2.92 

2.75 

2.61 

2.5 

2\33 

2.19 

2.06 

1.98 

1.84 

1.74 

1.57 

1.46 

30 

3.34 

3.11 

2.92 

2.78 

2.66 

2.47 

2.33 

2.21 

2.11 

1.945 

1.84 

1.68 

1.55 

35 

3.52 

3.27 

3.08 

2.93 

2.80 

2.60 

2.44 

2.33 

2.12 

2.06 

1.94 

1.76 

1.64 

40 

3.68 

3.42 

3.22 

3.06 

2.92 

2.71 

2.56 

2.43 

2.33 

2.15 

2.03 

1.85 

1.71 

45 

3.83 

3.56 

3.34 

3.18 

3.04 

2.82 

2.66 

2.53 

2.42 

2.24 

2.11 

1.92 

1.78 

50 

3.97 

3.69 

3.47 

3.29 

3.15 

2.92 

2.75 

2.62 

2.50 

2.33 

2.19 

1.99 

1.84 

60 

4.21 

3.91 

3.68 

3.50 

3.35 

3.11 

2.93 

2.78 

2.66 

2.47 

2.33 

2.11 

1.96 

70 

4.44 

4.12 

3.88 

3.69 

3.53 

3.27 

3.06 

2.93 

2.80 

2.60 

2.44 

2.22 

2.06 

80 

4.64 

4.31 

4.05 

3.86 

3.63 

3.42 

3.22 

3.06 

2.93 

2.71 

2.55 

2.33 

2.15 

90 

4.82 

4.48 

4.22 

4.01 

3.83 

3.56 

3.35 

3.18 

3.04 

2.82 

2.61 

2.42 

2.24 

100 

5.00 

4.64 

4.37 

4.15 

3.97 

3.68 

3.47 

3.29 

3.15 

2.92 

2.75 

2.50 

2.31 

120 

5.31 

4.94 

4.64 

4.41 

4.22 

3.92 

3.68 

3.50 

3.35 

3.11 

2.92 

2.66 

2.46 

140 

5.59 

5.20 

4.89 

4.64 

4.44 

4.12 

3.88 

3.68 

3.52 

3.27 

3.08 

2.80 

2.59 

160 

5.85 

5 . 4^3 

5.10 

4.85 

4.64 

4.31 

4.05 

3.85 

3.68 

3.42 

3.22 

2.92 

2.71 

180 

6.08 

5.65 

5.31 

5.05 

4.83 

4.48 

4.22 

4.01 

3.83 

3.56 

3.35 

3.04 

2.82 

200 

6.30 

5.85 

5.50 

5.23 

5.00 

4.64 

4.37 

4.15 

3.97 

3.68 

3.47 

3.15 

2.92 

250 

6.78 

6.30 

5.93 

5.51 

5.39 

5.00 

4.70 

4.47 

4.27 

3.97 

3.73 

3.39 

3.15 

300 

7.21 

6.69 

6.30 

6.00 

5.73 

5.31 

5.00 

4.75 

4.54 

4.11 

3.97 

3.61 

3.35 

350 

7.59 

7.05 

6.63 

6.30 

6.03 

5.59 

5.41 

5.00 

4.78 

4.44 

4.28 

3.80 

3.52 

400 

7.94 

7.37 

6.93 

6.59 

6.30 

5.85 

5.50 

5.23 

5.00 

4.64 

4.37 

3.92 

3.68 

450 

8.26 

7.66 

7.21 

6.85 

6.55 

6.08 

5.72 

5.44 

5.20 

4.83 

4.54 

4.13 

3.88 

500 

8.55 

7.94 

7.46 

7.10 

6.79 

6.30 

5.93 

5.63 

5.39 

5.00 

4.71 

4.27 

3.97 

550 

8.82 

8.19 

7.71 

7.32 

7.01 

6.50 

6.12 

5.81 

5.56 

5.16 

4.86 

4.41 

4.10 

600 

9.08 

8.43 

7.92 

7.54 

7.21 

6.69 

6.30 

6.00 

5.72 

5.32 

5.00 

4.54 

4.22 

700 

9.56 

8.88 

8.35 

7.94 

7.59 

7.05 

6.63 

6.30 

6.03 

5.59 

5.26 

4.78 

4.44 

800 

10.0 

9.28 

8.74 

8.30 

7.94 

7.37 

6.93 

6.59 

6.30 

5.85 

5.50 

5.00 

4.64 

1000 

10.8 

10.0 

9.41 

8.94 

8.55 

7.94 

7.47 

7.09 

6.79 

6.30 

5.93 

5.39 

5.00 

1200 

11.5 

10.6 

10.0 

9.50 

9.09 

8.43 

7.94 

7.54 

7.21 

6.69 

6.30 

5.72 

5.31 

1500 

12.3 

11.5 

10.7 

10.3 

9.79 

9.09 

8.55 

8.12 

7.77 

7.21 

6.79 

6.17 

5.72 

2000 

13.5 

12.6 

11.8 

11.2 

10.8 

10.0 

9.41 

8.94 

S .55 

7.94 

7.47 

6.79 

6.30 

2500 

14.6 

13.5 

12.8 

12.2 

11.6 

10.8 

10.2 

9.63 

9.21 

8.55 

8.05 

7.31 

6.73 

8000 

15.5 

14.4 

13.5 

12.9 

12.3 

11.4 

10.8 

10.2 

9.79 

9.09 

8.55 

7.77 

7.41 

3500 

16.4 

15.2 

14.3 

13.5 

13.0 

12.1 

11.3 

10.8 

10.3 

9.56 

9.00 

8.18 

7.59 

4000 

17.1 

15.9 

15.0 

14.2 

13.6 

12.6 

11.8 

11.3 

10.8 

10.0 

9.41 

8.55 

7.94 

4500 

17.8 

16.5 

15.5 

14.7 

14.1 

13.1 

12.3 

11.7 

1 1.2 

10.4 

9.79 

8.89 

8.25 

5000 

18.4 

17.1 

16.1 

15.3 

14.6 

13.5 

12 . 8 ! 12.1 

11.6 

11.8 

10.1 

9.21 

8.55 































{?LxOE Y ALVES. 


42!) 


SLIDE VALVES. 

The slide valve motion is one of the most important features in causing a 
steam engine to work well, and to employ the effect of steam economically. 
The author of this book being well acquainted with disarrangements on this 
point, has here endeavoured to give a good working-drawing of the proper pro¬ 
portions and arrangements of slide-valve motion. (See Plate IY.) 

Main Valve. 

It will he best to assume a certain size cylinder, and at the same time give the 
proportions for any size. 

D = 34 inches, diameter of the cylinder. 

S — 18 inches stroke of piston.* 
n = 56 double strokes per minute. 

We have the area of the steamports ra, from Formula 26, page 252, 


34«X0-7S5X18X56 


vi — 


30600 

D _±_ S 

26 


= 30 square inches, nearly. 


34+18 _ . , 

! —~~ = 2 inches, 


the width of the steamport; if the quotient gives a fraction take the nearest 
quarter or eighth. 

— = — = 15 inches, breadth of steamport. 

m 2 

r = i m about = 1 inch, the exhaust port o — 2m — = 3$ inches, and 

/ = o + 2r = 5i inches, h = f — |r = 5- inches, k = 1 = 3 inches, and 

i = h-\-2k = llj- inches, e = m — 2 inches. 

* The stroke and diameter is here rather out of proportion, but we will maintain 
them in the calculations as they suit the drawing, which is purposely made to 
show the slide vcdves on a large scale. The rules will however suit any propor¬ 
tions of diameter and stroke. 

To Find ilie Stroke off the Eccentric. 

5 = stroke of the eccentric in inches. 
s = i —/— \r = 5& inches. 

The lap L = \(i —/— 2m) = J inches. 

The lead of the valve, or opening of the steamport when the crank pin stands 
on the centre should he about 


l 


myn _2+56 

~80~ 80 


inches, nearly. 


Having finished the main valve and ascertained the stroke of the eccentric, 
it is now required to find the position of the ceutre b, (Plate V.,) of the eccentric, 
to the crank-pin. Suppose the crank pin of the engine stands atsa on the centre 
nearest to the cylinder, and the eccentric rods are attached direct to the valve 
rods; draw the line dd, at right-angle to the centre-line a a" of the engine, 
then 

the angle, sin. = 0-409, or W= 24° 10'. 

s 5 i 

See Plates IY. and Y. 

To Find tlve position off tlie Crank>Pin at tlie moment the 

Main Valve opens. 


y= 


s cos 


s l = J-gXO-25 

. W 5-5X0-9123 U y mcnes » nearly. 


from the centre line. 





























































































































f • - 












/ }/Z IP'S. 


Plate IF. 











































































































































































Erca/fr/cs 


Plate K 











































































. 





























■ 




- 













■ 
































. 












Slide Valves. 


421 


To Find tUe position of the Crank at tlie moment the 

Exhaust opens. 

x = ^sin.TF— ■!(/— *)) = y^O-409 — ^(5-5 — 5 ' 25 )) = 3 ' 27 incV,es 
from the centre line. 

To Find the position of the Crank Pin when the Main 
Valve cuts off the Steam* 

2 X 18 XJ 


X = 2 -^ = 


5-5 


5‘727 inches. 


To Find at what part of the Stroke the Main Valve Cuts 

off the Steam, 


Will cut off at = 1 


/ 2Xj\ a _ 

s2 \ 5-5 / 


0 - 899 of the stroke. 


The greater the lap is, the sooner will the main-valve cut off, hut if the lap is 
increased the stroke of the eccentric must also he equally increased. It does 
not work well to cut off much by the main-valve, especially when the engine 
works fast; for very slow motion it may answer to cut off at ft the stroke. 

It will he noticed that the centre of the eccentric is always ahead of the crank 
pin with an angle 90°Hence when the engine is to be reversed, the centre 
b must have the same position on the opposite side of the centre-line, or the 
eccentric must be moved forwards an angle of 90° — 2w. 

Cnt“Off Valve* 

The width of the cut off ports should be about d = = 1£ inch, and their 


x. a 30 

breadth 


= 12 inches, when two ports are used. 


Proportions of the Valve. 

a _ b — e _ d, a+d — b+c, and a = 2 d, and the stroke of the cut-off valve 

eccentric s = 2b, we shall have a — 2±, b = 2*, c = 1$, c — li, and 

5 LetViTassmne the steam to be cut off at f = l of the stroke S, the position of 
the crank-pin a' will then be sin.w = 21 = 0 - 666 , or u = 70 ° 30 ' ; at the same 
time the position of the centre d of the cut off eccentric will be 


sin 


= = li±li = 0-612, or 2 = 37° 50', 

8 4y 


and r= u — z == 70° 30' — 37° 50' 
the crank-pin a is on the centre, 
centre a and c, at different cut offs. 


= 32° 40', the position of the centre c when 
This Table will show the positions of the 
Letters correspond with Figure 1, Plate VI. 


Cut off 
at l. 


V 

sin.u 

stroke of 
eccen. s. 

22° 10' 

0-377 

2b 

32° 40' 

0-539 

2b 

31° 55' 

0-527 

c+a 

42° 35' 

0-675 

5+c 

46° 30' 

0-7193 

a+b — c 

50° 30' 

0-7933 

a-f-5— c 


t 

V 

F. 

P- 

37° 50' 

60° 

0-5880 

0-250 

37° 50' 

70° 30' 

0-6914 

0-333 

43° 35' 

75° 30' 

0-7332 

0-375 

47° 25' 

90° 

08350 

0-500 

58° 

104° 30' 

0-910 

0-625 

58° 30' 

109° 30' 

0-9S5 

0-666 


It will now be oDservm mat wie ... Tf fd , 

than in the Table on page 239, owing to the valve not cutting off the steam 
instantly, but gradually, so that the density of the steam in the cylinder is 
already diminished at the cut off point. The valve will cut off quicker the less 

th «ee n |igure "2, Plate VIII. The actual pressure will not form a sharp corner at 
e or follow the line e,e,e, as would be due when cut off at i the stroke, but the 
line ff'ff will be the true diagram. Including the steam in the ports and 
eteamchest, the density at the end of the stroke will correspond nearly with the 
Table. ___ 

























422 


Steam boilers. 


STEAM BOILERS. 

The accompanying proportions are averages of a great number of good 
marine boilers. 

Letters denote. 

D — diameter of the steam-cylinder in inches. 

S = stroke of piston under which steam is fully admitted, in inches. 
n = number of double strokes, or revolutions per minute. 
w = pounds of water evaporated per pound of coal, per hour. 

V = volum coefficient from the steam table. 

= fire grate in square feet, for each cylinder, and with natural draft. 

To Find tlie Area of Fire Grate* 


4*66 wV n ~~ I>* S ’ 


Example 1. A steam engine of D = 54 inches diameter of the cylinder, and 
stroke of piston 96 inches, cut off at £=48 inches; is to make 22 revolutions 
per minute. Anthracite coal to be used, that evaporates w = 7 pounds of 
water per pound of coal, and to carry 27 pounds of steam per square inch, 
V *= 649. Required the area of fire grate EE3 = ? in square feet. 


= 54*X48X22 
4*66X7X649 


145*34 square feet. 


"Example 2, A steamboiler of r=i = 128 square feet, is to be used for an 
engine of D = 36 inches diameter, and 64 inches stroke,—cut off the steam at 
4 then S — 42-66 inches. Steam pressure to be kept at 25 pounds per square 
inch V— 679. w == 6-5. Required for how many revolutions per minutes can 
the steam be kept at 25 pounds ? 


4-66X6-5X679X128 

<T7 _ __ 

36 2 X42-66 


= 47’6 revolutions. 


Horse Power of the Fire Grate. 

H = horse power of the fire grate. 

p = pressure in the boiler in pounds per square inch, excluding the 
atmosphere. 

vacuum in the condensor in pounds per square inch. 

Hx EEi Vw (P+ 0.8 p) . 

Vw (P+ 0.8^?) ’ x 


EE3 


3 4. 


Cut off the 
steam at 


fl‘ h 

I If 33 


J 1 

i 2 

I 9 


.? 


V 99 

99 


99 

99 

99 

99 


x = 27700. saves 55' 

x = 31400. 


49 

x = 38400. 


38 > 

x = 45500. 

33 

26 

x = 49100. 

33 

20 

x = 61700, 

33 

Op 


per cent 
of fuel. 


Steam admitted throughout the stroke x 

Example 3. Steamboilers are to be constructed for an engine of 650 horses, 
the steam to be cut off at J the stroke, and P= 36 pounds per square Inch, 
V— 544, w = 7-5 pounds of water evaporated per pound of coal. Required the 
fire grate in the boilers I-1 = ? in square feet. 


650 X 38400 
554 x 7.5(36 + 0.8 X 11)" 


136 square feet 



















Steam Boilers. 


*23 


Example 4. Required, the horse-power of a fire grate 13 = 112 square feet, to 
carry 18 pounds steam, and cut off at % the stroke ? V — 810, w = l pounds. 


H= 


112 X 18X810X7 
45500 


=•■ 251.2 horses. 


Coiisumi>tion of Coal. 

: coal consumed in pounds per hour. 


^ 3D 2 S n 

~~ w V ’ 


C- 


UHx 


Vw{P+0.8p) 


5, 6. 


Example 5. A steam-engine of D = 42 inches diameter, and 48 inches stroke, 
cut off the steam at ]/ s S= 16 inches, is to make n = 65 revolutions per minute with 
a pressure of 34 pounds per square inch, V = 564, and w = 6 pounds. Required, the 
consumption of coal in pounds per hour C= ? 

^ 3 X 42 2 X16 X 65 , v 

G— ——-—-—— = 1625 pounds per hour. 

6 X 564 

Example 6. A pair of steam-engines of 77=260 horses are to be worked with 
P = 28 pounds per square inch, cut off at X the stroke, V = 635, the coal to evap¬ 
orate w — 6.5 pounds of water per pound of coal. Required, the consumption of 
coal in pounds per hour C=l 


c= 


14 X 260 X 31400 
630 X 6.5(28 + 0.8 X 10) 


= 775 pounds per hour. 


It will be observed in the Formulas 4 and 6 that the higher steam used, the less 
fuel and fire-grate is required for the same power—the proportion of fuel will be 
nearly as the square root of the steam pressure, and still more fuel is saved by 
cutting off the steam at an early part of the stroke. 

Heating Surface O Compared with Grate. 

In common stationary boilers, . . O — 20|^jf* 

Returning flue boilers, . . . .0 = 25 1- 1. 

Tubular boilers (marine), . . Q = 30|^|. 

With vertical tubes (Martin), . . . Q = 35 |. 

Cross-area of Flues (Calorimeter). 

In the common single returning flue boilers, the cross-sectio n a rea of the first 

row should be,.0.18 13 

Returning row, flues or tubes, . . . 0.13 8 : I • 

Cross-section area of chimney at the top A = 0.16 fzzz^l - 


A=- 


C 2 


- 2 , 


Height of Chimney. 

jff 2 


4I3 2 

JI= 1.45^1/A, 


A = 


A = 


2.1 A 
JT 


1.45/A* 


C= 213/A+ 2, 
G 

13 


2/A+ 2 


Example. Area of fire-grate J=T = 140 sq. ft., to consume C= 2100 pounds of 
ccal per hour. Required, the height h of the chimney ? 


A = 


2100 2 


4 X 140 2 


— = 56.3 feet, the answer. 



















424 


Steam-Boilers. 


Standard Horse-Power of Steam-Boilers. 

The power of a steam-boiler ought to be graded by the dimensions of the areas 
of the fire grate and heating surface, like that of a steam-engine is graded by the 
diameter and stroke of the steam-piston, without taking into consideration the 
evaporative power of the fuel, expansion of the steam, etc., which are independent 
of the size of the boiler, as well as that of the engine. 

Let 1=1 denote the area of the fire grate. 

O = the area of the heating surface in square feet. 

P= pressure of steam in pounds per square inch above vacuum. 

Then the standard nominal horse-power H of a steam boiler can be expressed 
by— ) 


H ->/ 


OjAP. 


10 


F=> : 


10W 


o= 


10 W i 


o /P B/P 

Example. Suppose g= | = 100, 0 = 3000 and P= 


P= 


75, 


10 W \* 


ft 


30/ 


Then, 




yl 1 


510, the standard nominal 


100 x 3000 q/75 
10 

horse-power. 

Ordinary Performance of Steam-Boilers. 

Natural draft consumes about 12 to 15 pounds of coal per square foot of grate 
per hour, and generates about 4 to 5 horse-power per square foot of grate. 

The heating surface should be about 4 to 5 square feet per horse-power, and 
evaporate 4 to 5 pounds, or 92.5 to 115.5 cubic inches, of sea water pe* .tour, at the 
above-mentioned rate of combustion. 

Good coal evaporates about 6 to 8 pounds of water per pound of coal. 

Each horse-power requires the consumption of about 3 to 4 pounds of coal per 
hour. 

To find llie Ultimate Bursting Strength 

of a steam-boiler — shell, tube or flue. 

Notation of Letters. 

W= ultimate tensile strength in pounds per square inch of the boiler iron. 
t — thickness of the plate iron in decimals of an inch. 

D = diameter of the boiler in inches. 

P= bursting pressure (internal) in pounds per square inch. 

P= -?—> for single-riveted, and 


P= 


D 
l.ZWt 
D 


for double-riveted boilers. 


Example 1. The diameter of a single-riveted boiler being D = 72 inches, thick¬ 
ness of plates t = 0.375 inches, and the ultimate strength of the iron W = 4500C 
pounds to the square inch. Required, the bursting pressure of the boiler ? 

„ 45000x0.375 00 „ „ „ . „ 

P= -—-= 234.4 pounds to the square inch. 

A double-riveted boiler of the same dimensions will burst with 1.3 X 234.4 = 
304.72 pounds to the square inch. 

The Thickness of Boiler-Plates required for Bursting Pressure. 


t = 


PD 


W 

PD 


for single-riveted, and 


1.3 W 


for double-riveted boilers. 


























Steam-Boilers. 


m 


Ultimate Strength, of Tubes and Flues 

for External Pressure to Collapse. 

Notation of Letters. 

D — diameter of tube or flue in inches. 

L = length of the tube or flue in feet. 
t = thickness of iron in decimals of an inch. 

P= external collapsing pressure in pounds per square inch. 


P= 


200 . 000 1 2 


By' L 


and 


V PB j/2T 

-gfe- 


Example 1. A flue of D = 15 inches diameter, and L = 12 feet long, thickness 
of iron t = 0.25. Required, the collapsing pressure ? 

„ 200,000x0.25 2 , 

P=-—-=241 pounds to the square inch. 

15 x j/12 

Example 2. D = 9, L = 10 and t = 0.2. 

p= 200.000 x 0.04 g 282 pounds> 

9 |/10 

Example 3. P = 6, P = 6, and t = 0.2. Required the pressure P ? 

200,000x0.04 , 

P=- z - 73 — = 504 pounds. 

6 x -[/ 6 


Staying Steam-Boilers. 

cZ = diameter of good iron stay-bolts in inches. 

B = distance apart in inches in salt water on 
flat surfaces. 


ri,= B y r ~B 
74 ' 


B 


74 d 


P= pressure of steam in pounds per square inch. P= 


VP 

5476 d 2 
B 2 


The following table is given by Mr. Fairbairn, as exhibiting the strongest form 
and best proportions of rivet joints, as deduced from experiments and actual 
practice: 


Thickness 
of plate. 

Diameter of 
rivet. 

Length of rivet 
from head. 

Distance from 
centre to cent. 

Qnantity 
single riveted. 

of lap in 
double riveted. 

in. 16ths. 

in. 

Ratio. 

in. 

Ratio. 

in. 

Ratio. 

in. 

Ratio. 

in. 

Ratio. 

0.19= 3 

0.38 

2 

0.88 

4.5 

1.25 

6 

1.25 

6 

2.10 

10 

0.25= 4 

0.50 

2 

1.13 

4.5 

1.50 

6 

1.50 

6 

2.50 

10 

0.31= 5 

0.63 

2 

1.38 

4.5 

1.63 

5 

1.88 

6 

3.15 

10 

0.38= 6 

0.75 

2 

1.63 

4.5 

1.75 

5 

2.00 

5.5 

3.33 

9.2 

0.50= 8 

0.81 

1.5 

2.25 

4.5 

2.00 

4 

2.25 

4.5 

3.75 

7.5 

0.63 = 10 

0.94 

1.5 

2.75 

4.5 

2.50 

4 

2.75 

4.5 

4.58 

7.5 

0.75 = 12 

1.13 

1.5 

3.25 

4.5 

3.00 

4 

3.25 

4.5 

5.42 

7.5 





































426 


Fuel and Timber. 


WOOD FOR COMBUSTION. 

A cord of wood is 8 feet wide by 4 feet high and 4 feet deep, or the wood is 4 feet 
loug. The cord contains 8 X 4 X 4 = 128 cubic feet, of which only 74 cubic feet 
is solid wood and 54 cubic feet of space. 

Two cords of wood evaporate about the same quantity of water as one ton of 
anthracite coal. 

The best pine wood evaporates 5 pouuds of water per pound of wood consumed 
in a steam-boiler furnace. One cord of wood can be consumed per hour on 60 square 
feet of grate. 

“Weight in Pounds per Cord of Different Woods. 


Woods, Seasoned. 

lbs. 

Woods, Seasoned. 

lbs. 

Woods, Seasoned. 

lbs. 

Shell-bark Hickory. 

4469 

Hard Maple . . . 

2878 

Cedar ..... 

1910 

White Oak.... 

3821 

Beech . 

2875 

Yellow Pine . . . 

1904 

Ked-heart Hickory . 

3705 

Hazel. 

2870! 

White Pine . . . 

1868 

Southern Pine . . 

3375 

Virginia Pine . . 

2689 

Spruce . 

1685 

Red Oak .... 

3254 

New Jersey Pine • . 

2137 

Hemlock .... 

1240 


Wood requires 32 per cent, more fire grate than mineral coal, for equal genera¬ 
tion of steam. The furnace should be 60 per cent, of cubical space more for wood 
than for coal, or about 4.5 cubic feet per square foot of grate. 

Properties of Fuel. 


Kind of Fuel. 


Bituminous coal 
Anthracite coal . 

Coke .... 
Coke, n at. Virginia . 

Coke, Cumberland 
Charcoal 
Dry wood . 

Wood with 20 per ct. water 
Turf, dry . 

Turf, 20 per ct. water 
Oil, Wax, Tallow 
Alcohol (from market) 
Chemically, one pound of 
153 


Units of heat 
per pound of 
fuel. 

Pounds of water 
evaporated per 
lb. of coal. 

Per cent, of 
carbon. 

Cubic feet of air 
requ. for one lb. 
of coal. 

Weight per 
cubic foot. 

Cubic feet to 

stow a ton. 

11600 

7 to 9 

80 

'265 

50 

44 

13340 

8 to 10 

92 

282 

54 

40 

12420 

8 to 10 

86 

245 

31 

72 

11600 

8 to 9 

80 

260 

48 

48 

11600 

8 to 10 

80 

250 

32 

70 

13920 

5 to 6 

96 

265 

24 

104 

6380 

4 to 5 

44 

147 

20 

100 

4930 

4 

34 

115 

25 

100 

7395 

6 

51 

165 

28 

8n 

6800 

5 

40 

132 

30 

75 

11165 

14 

77 

200 

59 

37 

8410 

9.56 

58 

154 

52 

42 


carbon burnt 
cubic feet of 


to carbonic acid 
atmospheric air. 


requires the oxygen of 


Timber, Green and Seasoned. 


Timber. 

Green . 

Seasoned. 

American Pine. 


44.75 

30.69 

Ash • • • • 


58.19 

50.00 

Beech 


60.00 

53.37 

Cedar .... 


32.00 

28.25 

English Oak 


71.62 

43.50 

Riga Fir 


48.75 

35.50 




Comparative weight per cu- 
V bic foot in pounds of green 
( and seasoned timber. 




Board Measure. 

Multiply together the three dimensions, width and thickness in inches and the 
length of the lumber in feet; divide the product by 12, and the quotient will be 
the board measure. 














































Combustion and Effect ©f Fuel 


427 


Combustion is the rapid chemical combination of substances with 
oxygen. Carbon C and hydrogen H, are the substances most generally 
employed for generating heat. Carbon is fully consumed when combined 
with oxygen 0, to form carbonic acid gas COi , and partly consumed 
when in the form of carbonic oxide gas CO or smoke. /i = units of heat 
generated of one pound of fuel. The heat necessary to raise one pound 
of water one degi*ee Fall, is one unit of heat. w=pounds of water at 
212 ° evaporated per pound of fuel. 2l=volume in cubic feet and «=weight 
in pounds of atmospheric air required for the perfect combustion of one 
pound of fuel. C, 0, and H are in the four first formulas, fractions in 
one pound of the compound fuel. 


Perfect, Combustion. 
a=149[C+3(H-|)],- - 1 

a=12[C+3(fl-f)], - - 2 

7 v =14500[CH-4-28(fl—|)X 3 

15 [0+4*28(1?—^)] 


w = 


966 


Imperfect Combustion. 
(00)= 66C '- 210 


(co a y. 


Y2 

33 0- 44 C 
“12 ’ 


- 5 


- 6 


7t=3960(<7<9,)+1650( (7(9), - 7 
7i=8002-5(9—6820(7, - - 8 


When oxygen is supplied to carbon in a proportion between CO and 
COi, both the gases will be formed separately in the proportion of the 
formulas 5 and 6, when the heat generated will be as formulas 7 and 8, in 
which C, 0, CO and COi are expressed in pounds, for instance : 0= 20 lbs. 
of oxygen united with C=12 lbs. of carbon will form 


( 00 )=' 


56X12—21X20 


(C0 2 )— 


12 

33X20—44X12 


=21 lbs. carbonic oxide. 


12 


=11 lbs. carbonic acid, and will 


generate 7i=8002-5X20—6820X12=78210 units of heat. 

One unit of heat=772 foot pounds, if generated per second will be H= 
1.4 horses, of which we in our days practice utilizes about one-twentieth. 
The following table will show how important it is to fully consume the 
combustibles to acid. One pound of carbon consumed to oxide will gen¬ 
erate only T72 horses, instead of 6 66 when consumed to acid. 

Properties of Combustion, per Hour. 


c 

CO 

C0 2 

O 

a 

A 

h 

w 

R 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

cub. ft. 

heat. 

lbs. 

horses. 

1 


3*666 

2*666 

12 

149 

14500 

15 

5*660 

1 

2*666 


1*333 

6 

74*50 

4400 

4*55 

1*720 

0*433 

1 


0*566 

2*550 

31*65 

1650 

5*63? 

1*200 

0*272 


1 

0*727 

3*275 

40*56 

3960 

4*100 

1*545 


1*750 

1*375 

1 

3*500 

43*33 

5440 

5*633 

2*125 


0*445 

0*392 

0.222 

1 

12*38 

1210 

1*250 

0*472 


•0358 

•0246 

•0211 

•0808 

f 

97*3 

0*100 

0*038 


0*584 

0*244 

0*170 

0*800 

9*920 

966 

1 

0-37S 


1*550 

0*651 

0*470 

2*120 

26*30 

2558 

2*645 

a 




































428 


Radiation' of Heat from Steam-Pipes. 


RADIATION OF HEAT FROM STEAM-PIPES, 

Boilers or Steam Cylinders. 

Notation of Letters. 

D = outside diameter of steam-pipe, without casing, and limited to not more 
than 12 inches. 

T = temperature of the steam, Fahr. scale. 
t = temperature of the external air. 

h = calorics radiated per square foot per hour, on uncovered pipe. 

A — outside area in square feet of steam-pipe. 

H — horse-power lost by radiation of heat. 



Wind. 

Exp. n. 

n = exponent of the wind, which varies with the cur- 

Calm. 

1.20 

rent of air or draft about the steam-pipe, as in 

Gentle. 

1.22 

the following table: 

Brisk. 

1.24 


Storm. 

1.26 

The loss of heat will then be per hour— 



h = 0.001122 [450 + (12 — D) 7 ] ( T—t) n 

• • 

. 1. 

Horse-power lost IT — jAJL- . 

2564 

• • • 

2. 

One horse-power consumes or generates 2564 calorics per hour. 

By logarithms the Formula 1 is reduced to— 


log. h = log. Jc -f n log. (T — t). . 

• • 

. 3. 

The log. 7c is contained in the second column of the accompanying table for 
different diameters of pipes. 

For any uncovered plane or cylindrical surface above 12 inches in diameter the 
radiation in units of heat per square foot per hour will be— 

h = 0.505 (T — t) n . 

• • • 

4. 


The effect of thickness of metal is inappreciable for practical purposes. 

Example 1. The California S. N. Co.’s steamer Julia has a steam-pipe 40 feet 
long by D = 9 inches in diameter, and two branch-pipes 12 feet long by D = 4 
inches each, all uncovered. Pressure of steam, 100 lbs. T = 337°. Temperature 
of the external air t = 70°. Required, the loss of heat and power by radiation? 

In calm wind n = 1.2. See table for n. 

h = 0.001122 [450 + (12 — 9) 2 ] (337° — 70 0 ) 1 - 2 = 420.34, 
the units of heat lost per square foot per hour. 

Area of pipe, A = 0.75 X 3.14 X 40 = 94.24 square feet. 

Power lost, H = 94 ' 24 X 4204 = 15.5 horses. 

2564 

The branch-pipes lose 4.6 horses. 

The total loss of power 20.1 horses. 

The same pipes covered with 2-inch-thick felt would gain 20.1 X 0.93 = 18.7 
horse-power. 

Example 2. In the factory of Bellavista, Peru, are 150 feet of uncovered steam- 
pipes D = 3 inches in diameter. Steam-pressure, 45 lbs. 2’= 292° Fahr. 
Temperature of external air £ = 68, and wind gentle, n = 1.22. Required, the 
horse-power and fuel lost? 

Formula 3. log. h = 0.77408 — 1 +1.22 log. (292 — 68) = 2.32805, 
or 173 units of heat lost per hour. 

Area of steam-pipes, A = 0.785 X 150 = 117.8 square feet. 

1J7 8 V 173 

Formula 2. H = — 1 —^-= 8 horse-power nearly, which is lost by radiation. 

2564: 











Radiation of Heat from Steam-Pipes, 


429 


The same pipes covered with one-inch felt will gain 8 X 0.89 (see table) = 7.12 
horse-power. The steam-engine in Bella vista works without expansion, and con¬ 
sumes about 10 Bbs. of coal per horse-power per hour = 7.12 X 10 = 71.2 pounds, 
and for 8 hours’ working = 569.6 lbs. of coal lost per day. 

The radiation of heat from steam-pipes causes a condensation of steam to water, 
and the weight iir pounds of water 60 condensed is equal to the units of heat 
radiated, divided by the latent heat of the steam in the pipe. The Formula 1 will 
also answer for calculating the quantity of heat radiated from steam or water- 
pipes for heating rooms. 


Percentage of Heat or Power Gained 

by covering steam-pipes with felt and canvas outside. 


Diam 

D 

Logarithm 

k. 

l 

¥ 

3 

8 

Thick 

1 

2 

ness o 

3 

4 

f felt 

1 

joverii 

H 

ig in i 
2 

nches. 

3 

4 

6 

1 

0.80663—1 

65 

76 

81 

86 

92 

94 

96 

98 

99 

100 

2 

0.79561—1 

63' 

74 

80 

85 

90 

93 

95 

97 

98 

99 

3 

0.77408—1 

61 

72 

79 

84 

89 

92 

95 

96 

98 

99 

4 

0.76096—1 

59 

71 

77 

83 

83 

92 

94 

96 

97 

99 

5 

0.74809—1 

57 

69 

76 

82 

87 

91 

94 

96 

97 

99 

6 

0.73670—1 

54 

67 

74 

81 

86 

91 

94 

95 

97 

99 

7 

0.72668—1 

52 

66 

73 

81 

85 

90 

93 

95 

97 

99 

8 

0.71838—1 

50 

64 

71 

80 

85 

90 

• 93 

95 

97 

99 

9 

0.71179—1 

47 

62 

70 

79 

85 

89 

93 

95 

97 

99 

10 

0.70705—1 

45 

61 

69 

78 

84 

89 

92 

95 

96 

99 

11 

0.70417—1 

42 

59 

67 

78 

83 

88 

92 

94 

96 

98 

12 

0.70321—1 

40 

58 

66 

77 

83 

88 

92 

94 

96 

98 


Lap-welded American Charcoal Iron Boiler Tubes. 

Pascal Iron Works Tasker Iron Works, 

Philadelphia. New Castle, Del. 


Diameter of 

Heating surface 


Length of tube 

Area of cross - 

Weight 

the tube. 

per foot of length. 

of metal. 

per square foot . 

section . 

per foot of 

Outside 

Inside. 

Outside 

Inside. 

Outside. 

Inside . 

Outside 

Inside . 

length. 

Inches. 

Inches. 

Sq. ft. 

Sq. ft. 

Wg. inches. 

Feet. 

Feet. 

Sq. in. 

Sq.in. 

Pounds. 

1 

0.856 

0.2618 

0.2241 

15 0.072 

3.819 

4.460 

0.785 

0.575 

0.708 

1.25 

1.106 

0.3272 

0.2895 

15 0.072 

3.056 

3.455 

1.227 

0.960 

0.9 

1.50 

1.334 

0.3926 

0.3492 

14 0.083 

2.547 

2.863 

1.767 

1.396 

1.250 

1.75 

1.560 

0.4580 

0.4084 

13 0.095 

2.183 

2.448 

2.405 

1.911 

1.665 

2 

1.804 

0.5236 

0.4723 

13 0 098 

1.909 

2.118 

3.142 

2.556 

1.981 

2.25 

2.054 

0.5890 

0.5377 

13 0.098 

1.698 

1.850 

3.976 

3.314 

2.238 

2.50 

2.283 

0.6545 

0.5977 

12 0.109 

1.528 

1 673 

4.909 

4.094 

2.755 

2.75 

2.533 

0.7200 

0.6631 

13 0.109 

1.390 

1.508 

5.940 

5.039 

3.045 

3 

2.783 

0.7853 

0 7285 

12 0.109 

1.273 

1.373 

7.069 

6 083 

3.333 

3.25 

3.012 

0.8508 

0.7885 

11 0.119 

1.175 

1.268 

8.296 

7 125 

3.958 

3.50 

3.262 

0.9163 

0.8430 

11 0.119 

1.091 

1.171 

9.62 L 

8.357 

4.272 

3.75 

3.512 

0.9817 

0.9194 

11 0.119 

1.018 

1.088 

11.045 

9.687 

4.590 

4 

3.741 

1.0472 

0.9794 

10 0.130 

0.955 

1.023 

12.566 

10.992 

5.320 

4.50 

4.241 

1.1781 

1.1105 

10 0.130 

0.849 

0.901 

15.904 

14.126 

6.010 

5 

4.720 

1.3680 

1.2357 

9.5 0.140 

0.764 

0.809 

19.635 

17 497 

7.226 

6 

5.699 

1.5708 

1.4920 

9 0.151 

0.637 

0.670 

28.274 

25.509 

9.348 

7 

6657 

1.8326 

1.7428 

7.5 0.172 

0.545 

0.574 

38.484 

34.805 

12.435 

8 

7.036 

2.0944 

1.9991 

7 0.182 

0.478 

0.500 

50.265 

45.795 

15.109 

9 

8.615 

2.3562 

2.2553 

6.5 0.193 

0.424 

0.444 

63.617 

58.291 

18.002 

10 

9.573 

2.5347 

2.5022 

5.5 0.214 

0.382 

0 399 

78.540 

71.975 

22.19 


The length of tube and thickness of metal can be varied to suit orders. 

The heating surface of a. boiler tube is that exposed to the fire. 

Safe ends of thicker metal welded on the ends of tubes as may be required. 


1 



















































Blowing off. Incrustation. 


430 


BLOWING OFF. SALTWATER. INCRUSTATION. 

Sea water contains about 0.03 its weight of salt. When salt water boils, fresh 
iwater evaporates and the salt remains in the boiler; consequently the proportion 
of salt increases as the water evaporates, until it has reached 0.36 weight to the 
water; the salt will then commence to saturate in the boiler, and the water in so¬ 
lution will hold 0.36 weight of salt to 1 of water. 

To prevent deposit in the boiler, it is necessary to keep the salt below this pro¬ 
portion, which is overcome by withdrawing (blowing off) part of the supersalted 
water, while less salted (feed) water is replaced. It is found in practice that when 
the proportions are kept 0.12 of salt to 1 weight of Avater, the deposit will be very 
slight. To obtain this it will be necessary to blow off— 


(L03 

0.12 


= 0.25 parts of the feed water, or, 


if a brine-pump is used, it should be at least 0.25 of the feed-pump. 
W— cubic feet of supersalted water to be blown off per minute. 
Z>, S, n and F, as before, we shall have— 


w — 


P*S7l 

3000 F* 


Example. 2) = 30 inches, stroke of piston 36 inches, cut off at half stroke S= 
18, making 14 revolutions per minute, with a pressure of 30 pounds per square 
inch, F= 610. How much water must be blown off per minute? 

TF= 30 2 X 18 X 14 __ qj 24 cubic feet. 

3000 X 610 

Heat Wasted l>y Blowing Off. 

• Letters denote , 

re = water evaporated j j bi f t unit of tirae . 
and W= water blown off J F 

t = temperature of the feed water. 

T— “ “ blowing off. 

H— heat wasted, per cent. 

jj- W(T-t) 

^(990 + T — t) 

Example. Let the quantity of water blown off be % of the feed water, we have 
IF = 1, and w = 2; the boiling-point of the water will then be T= 215.5°; let the 
feed water taken from the hot-well be t = 100°. Required, the quantity of heat 
lost? 

1(215.50-100) = 0.052 or 5.2 per cent. 


H= 


2(990 + 215.5 — 100) 

This is a very trifling quantity of heat lost. 

Heat Wasted toy Incrustation.. 

The conducting power of iron for heat.is about 30 times that of saturated scales, 
hence a considerable portion of heat is lost when the scales become thick in a 
boiler. 


t = thickness of the scale in 16ths of an inch. ^ 

H— per cent, of heat wasted. 32 p " 

Example. The scale in a boiler is 5-sixteenths of an inch thick. How much 
heat is lost by it? 

5 2 




0.438, or 44 per cent., nearly, 


32 + 52 

which goes out through the chimney. 

This is merely to show that the heat lost by blowing off is but trifling compared 
with the heat lost by saturation of scales, Avh'ich additionally injures the boiler by 
softening and fracturing the iron, and final explosions. 

When boilers are fcdcen good care of by cleaning and blowing off at short inter¬ 
vals, the scales need not exceed 1-sixteenth of an inch. 















Belt, Signals. 


431 


Proportions of Salt in. Waters 

its boiling-point and iveight per cubic foot. 


Salt 

Boiling 

Weight 

Spe- 

Salt 

Boiling 

Weight 

Spe- 

in 100 

temp. 

per 

cific 

in 100 

temp. 

per 

cific 

Weights. 

Fahr. 

cub. ft. 

grav. 

weights. 

Fahr. 

cub. ft. 

grav. 



pounds. 




pounds. 


0 

212° 

59.837 

1.00 

21 

218.804 

72.224 

1.21 

1 

212.205 

60.431 

1.01 

22 

218.690 

72.728 

1.22 

2 

212.422 

61.024 

1.02 

23 

219.082 

73.395 

1.23 

3 

212.619 

61.617 

1.03 

24 

219.483 

73.980 

1.24 

4 

212.887 

62.209 

1.04 

25 

219.887 

74.565 

1.25 

5 

213.136 

62 801 

1.05 

26 

220.296 

75.148 

1.26 

6 

213.294 

63.393 

1.06 

27 

220.713 

75.732 

1.27 

7 

213.664 

63.984 

1.07 

28 

221.131 

76.316 

1.28 

8 

213.942 

64.575 

1.08 

29 

221.558 

76.899 

1.29 

9 

214.229 

65.166 

1.09 

30 

221.984 

77.482 

1.30 

10 

214.526 

65.756 

1.10 

31 

222.419 

78.064 

1.31 

11 

214.801 

66.346 

1.11 

32 

222.857 

78.646 

1.32 

12 

215.145 

66.935 

1.12 

33 

223 302 

79.228 

1.33 

13 

215.446 

67.524 

1.13 

34 

223.733 

79.810 

134 

14 

215.797 

68.113 

1.14 

35 

221.208 

80.390 

1.35 

15 

216.132 

63.701 

1.15 

36 

224.668 

80.970 

1.36 

16 

216.477 

69.289 

1.16 

37 

225.139 

81.550 

1.37 

17 

216.826 

69.877 

1.17 

38 

225.611 

82.130 

1.38 

18 

217.186 

70.464 

1.18 

39 

226.087 

82.709 

1.39 

19 

217.550 

71 051 

1.19 

40 

226.572 

83.288 

1.40 

20 

217.924 

71.377 

1.20 

Saturates with 40 parts of salt. 

Water 

does not increase in volume by addition of the above proportions of salt. 

To Command tlie Engineer liow to Manoeuvre tlie Engine in a 




Steamboat. 




Go ahead, . 

• i 

• 

• • 

one stroke. 


Back, 

« • 

• J 

-i 

• 

. two strokes. 


Stop, 

• • 

• J 

• 

• • 

one stroke. 


Slowly, 

• • 

• 

"I • 

• 

. two short. 


Full speed, 

Ml 

-dp- -<&- 

• • 

three short. 










Go ahead slowly, 

JJ 

• 

. one long, two short. 

j Back slowly, 


i n 

—tf£P— 

• 0 

two long, two short. 

Go ahead, full speed, _ J_ _<J- . J_ _*J. 

• 

. one long, three short 

Back fast, . 

• -«!- 

ITT! . 

two long, three short 

Hurry, 

, # 

. j 

1 1 1 

-tS 

1 1 1 

>- -m- » 

. three short repeated. 


It is also customary to have two hells in the engine-room—a large bell i'or the 
long strokes, and a smaller for the short strokes. 




































132 Brass Tubes, Strength op Iron and Copper. 


Weight, Size, Price and Surface of Copper and Brass Tubes, 

10 feet long. 


Outside 

Diam¬ 

eter. 

Bir.W. 

Gauge 

Weight 

Brass. 

of tube. 

Cop. 

Price 

per 

tube. 

Whole 

Sur¬ 

face. 

Outside 

Diam¬ 

eter. 

Bir.W. 

Gauge 

Weight 

Brass. 

of tube. 

Cop. 

Price 

per 

tube. 

Whole 

Sur¬ 

face. 

Inches. 

No. 

Lbs. 

Lbs. 

$ cts. 

Sq. Ft. 

Inches. 

No. 

Lbs. 

Lbs. 

$ cts. 

Sq. Ft. 

0.625 

18 

3.478 

3,681 

2 30 

1.636 

2. 

14 

18.84 

19.95 

8 00 

5.236 

0.75 

17 

4.950 

5.241 

2 97 

1.963 

2.125 

14 

19.07 

20.18 

8 20 

5.563 

.8125 

17 

5.372 

5.679 

3 00 

2.127 

2.25 

14 

21.18 

22.42 

8 90 

5.890 

0.875 

17 

5.775 

6.114 

3 40 

2.290 

3.375 

14 

22.32 

23.65 

9 45 

6.217 

.9375 

16 

6.954 

7.362 

3 60 

2.454 

2.5 

14 

23.53 

24.89 

9 95 

6.544 

1 . 

16 

7.418 

5.854 

3 92 

2.618 

2.625 

14 

24.67 

26.12 

10 45 

6.872 

1.125 

16 

8.354 

8.835 

4 40 

2.945 

2.75 

14 

25.83 

27.35 

11 00 

7.200 

1.25 

15 

10.21 

10.83 

4 70 

3.272 

3. 

13 

37.00 

39.17 

13 70 

7.854 

1.375 

15 

11.23 

11.91 

4 95 

3.000 

3.25 

13 

40.00 

42.34 

14 85 

8.508 

1.5 

15 

12.28 

13.00 

5 20 

3.927 

3.5 

13 

43.10 

45.61 

16 00 

9.103 

1.625 

15 

13.30 

14.08 

5 65 

4.254 

4. 

12 

49.39 

52.30 

17 00 

10.47 

1.75 

14 

16.5 

17.45 

7 00 

4.581 

4.5 

12 

55.55 

58.8 

19 10 

11.78 

1.812 

14 

17.08 

18.08 

7 20 

4.745 

5. 

12 

61.44 

65.00 

21 00 

13.08 

1.875 

14 

17.72 

18.75 

7 50 

4.908 

6. 

11 

81.58 

86.35 

26 00 

15.71 

1.937 

14 

18.26 

19.32 

7 75 

5.072 

8. 

11 

108.8 

115.0 

34 50 

20.95 


Scamless-Drawn Brass Tubes for Plumbing, 

In lengths of 10 feet. Screw-coupling on one end of each length. Price per tube. 


Diameters, inches. 

5 

8 

3 

4 

1 

1 


n 

Plain tubes, . 

Tinned tubes, . 

$ cts. 

2 50 

3 00 

$ cts. 

3 00 

3 50 

$ cts. 

4 50 

5 00 

$ cts. 

6 00 

7 00 

$ cts. 

7 00 

8 00 

$ cts. 

8 00 

9 00 

Price of Taps, Dies and Stocks. 

Diameters, inches. 

5 

8 

3 

¥ 

7 

8 

1 

11 


Taps, .... 
Solid dies, 

$ cts. 

2 50 

3 50 

St 

$ cts. 

2 75 

3 50 
ocks, $8, 

$ cts. 

3 00 

3 50 
□et. 

$ cts. 

3 50 

3 50 

$ cts. 

4 50 

3 50 

$ cts. 

6 00 

4 00 

Price for Each Extra Coupling. 

Diameters, inches. 

0 5 

8 

3 ' 

4 

1 

. ± . 

H* 

n 

Straight couplings, 
Elbows, 

Tees, .... 
Cross couplings, 

$ cts. 

20 

26 

* 30 “ 
45 

$ cts. 3 
25 

35 

40 

60 

$ cts. 

35 

48 

55 

85 

$ cts. 

40 

53 

* 60 

90 

Hi Cts. 

45 

65 

85 

1 50 

$ cts. 

50 

80 

1 00 

2 20 


The prices are only approximate. The price of copper tubes is 12 to 13 per cent, 
more than of brass. 

Brass and copper tubes are manufactured at the American Tube Works, Boston, 
Mass.; Merchant & Co., 507 Market street, Philadelphia, agents. 


Proportionate Tensile Strength of Rolled Iron and Copper, 

In pounds per square inch, at different temperatures, Fahr. and Centigrade. 


Fahr. 

Cent. 

Iron. 

Copper. 

Fahr. 

Cent. 

Iron. 

Copper. 

32 

0. 

55,000 

32.800 

800 

427 

51,800 

17,200 

100 

37.7 

58,200 

32,300 

900 

483 

45,000 

14,000 

200 

93.3 

62,800 

31,000 

1000 

540 

37,000 

11,000 

300 

149. 

65,750 

29,500 

1200 

650 

25,000 

7,000 

400 

205. 

67,000 

27,400 

1500 

820 

16,500 

3,000 

500 

260. 

66,000 

25,300 

2000 

1090 

7.000 

0.0000 

600 

316. 

62,700 

23,000 

2500 

1370 

2,500 

Fused to 

700 

370. 

57,800 

20,100 

3000 

1650 

Fused. 

liquid. 























































































■Nails, Rivets, Iron, Copper, Zinc. 


433 


Composition Nails, Copper and Iron Rivets. 


oO 

Composition Nails. 

Braziers’ Copper Rivets. 

O 

fco 

Iron Rivets. 


6 

Thick. 

Length. 

In 1 

n>. 

Diameter. 

Length. 

In 10 

lbs. 

1 

o 

Diameter. 

Length.’ 

In 10 
lbs. 

No. 

Iuehes. 

Inches. 

Num 

Inches. 

Inches. 

Num. 

No. 

Inches. 

Inches. 

Num. 

1 

0.04 

3/4 

290 

3/16 

1/2 

2384 

0 

3/16 

1/2 

3280 

2 

0.05 

7/8 

260 

1/4 

1/2 

1018 

1 

1 /4 

1 / 2 

1276 

3 

0.06 

1 inch. 

212 

174 

9/16 

983 

2 

1/4 

9/16 

1130 

4 

0.07 

1.1/8 

201 

5/16 

9/16 

573 

3 

5/16 

9/16 

654 

5 

0.08 

1.1/4 

199 

5/16 

5/8 

516 

4 

5/16 

5/8 

589 

6 

0.09 

1 inch. 

190 

3/8 

7/8 

357 

5 

3/8 

7/8 

407 

7 

0.10 

1.1/8 

184 

3 / 8 

15/16 

334 

6 

3/8 

15/16 

380 

8 

0.10 

1.1/4 

168 

7 /16 

1 inch. 

210 

7 

7/16 

1 inch. 

239 

9 

0.11 

1.1 /2 

110 

1/2 

1.3 /16 

141 

8 

1/2 

1.3/16 

160 

10 

0.11 

1.5 /8 

101 

9 /16 

1.5/16 

99.5 

9 

9/16 

1.5/16 

112 

11 

0.12 

1.3/4 

74 

5/8 

1.7/16 

71.9 

10 

5/8 

1.7 /'16 

81.7 

12 

0.12 

2 inches 

64 

11/16 

1.9/16 

53.8 

11 

11 /16 

1.9/16 

61.3 

13 

0.13 

2.1 /4 

59 

3/4 

1.3 / 4 

41.6 

12 

3/4 

1.3 /4 

47.3 

14 

0.14 

2.1/2 

51 

13 /16 

1.13/16 

32.8 

13 

13/16 

1.13 /'16 

37.3 

15 

0.15 

2.3 f 4 

43 

7/8 

2.1 /16 

26.3 

14 

7/8 

2.1 /16 

30. 

16 

0.16 

3 inches 

35 

1 inch. 

2.3/8 

16.7 

15 

1 inch. 

2.3/8. 

19. 


Length in Indies of Penny Nails. 


1 in. 

1.25 

1.5 

1.75 

,2 

2.25 

2.5 

2.75 

3 

3.25 

3.5 

4 

4.25 

5 

5.5 

6 

2d. 

3 d. 

id. 

5 d. 

6 d. 

7 d. 

8 d. 

9 d. 

10 

12 

16 

20 

30 

40 

50 

60 


Sheet Zinc and Iron. 


Sheet Zinc.. 

Size 84 in. by 24, 28, 32, 36 and 40 inches. 

Russia Sheet Iron. 

Size 28 X 56 in. = 10.88 sq. feet. 

Zinc 

Width of Sheet. 

Bir. W. 

Russian 

Weight per 

Bir. W 

gauge. 

2d 

32 

40 

gauge. 

gauge. 

Sheet. 

Sq. Ft. 

gauge. 

No. 

Pounds. 

Pounds. 

Pounds. 

No. 

No. 

Pounds. 

Pounds. 

No. - 

8 

6.23 

9.68 

12.1 

28 

7 

6.25 

0.574 

29 

9 

7.20 

11.2 

14.0 

27 

8 

7.25 

0.666 

28 

10 

8.00 

12 4 

15.6 

26 

9 

8. 

0.735 

27 

11 

8.90 

13.8 

17.3 

25 

10 

9. 

0.S27 

26 

12 

10.1 

15.7 

19.7 

24 5 

11 

10. 

0.918 

25 

13 

11.1 

17.3 

21.6 

23 

12 

10.75 

0.987 

24.1 

14 

12.4 

19.3 

24.1 

22 

13 

11.75 

1.08 

24 

15 

16.2 

25.2 

31.6 

21 

14 

12.5 

1.15 

23J 

16 

17.4 

27.1 

33.9 

20 

15 

13.5 

1.24 

22 f 

18 

21.9 

34.0 

42.6 

18 

16 

14.5 

1.33 

2 H 


the weight 
of the Pattern by 


To find tile Weiglit of Castings, toy tlie Weight of Pine Patterns. 

RULE.— f 12 for Cast Iron, \ 

Multiply the weight J f and the product is the 

12.2 for Tin, f weight of the Castings. 

11.4 for Zinc, J 

Reductions for Round Cores and Core-prints. 

Rule. Multiply the square of the diameter by the length of the Core in 
inches, and the product by 0.017, is the weight of the pine core, to be deducted 
from the weight of the pattern. 

Shrinking of Castings. 


Pattern-Makers' Rule 
should be for 


Cast Iron, 

l 

• 8 

Brass, . 

3 

• 1 6 

Lead, . . 

1 

• 8 

Tin, . . 

1 

• TJ 

Zinc, . . 

3 

• 1 6 


of an inch longer per 
linear foot. 


28 










































































434 


Weight of Boilebs and Engines. 


To Approximate the Weight of Steam Boilers. 

The area of Are grate gives a nearer approximation to the weight ®f 
Marine boilers, than the heating surface. 

Letters denote. 

5 ==j = total fire grate in square feet. 

W = weight of the boiler in pounds, including fire bars, doors, smoke 
pipe, fire tools and appendages, but without water. fF=800 E3 • 

Example. Required the weight W=1 of a steam boiler of E3=250 
square feet, grate surface. 

W=800X250=200,000 lbs. 

Weight of the water is about three-fourths of W or of the total weight 
of boilers. 

Weight of rivets, braces or stays, doors and fire bars, is about one 
quarter of W or of the total weight of boilers. 

To Approximate the Weight of Engines. 

Letters denote 

^ — strobe^} c y^ n ^ er inches. 

W= weight of engine in pounds, including engine room tools, oil and 
tallow tanks, wheels, propeller and shafts. 


coefficient k. 

Trunk and oscillating engines, -------4 

Direct action paddle wheel engines, ------ 4-25 

Horizontal direct action propeller engine, - - - - 4-5 

Geared propeller engines, - -- -- -- - 5* 

American overhead beam engines, ------ 6-5 

Side lever engines, - -- -- -- -- - 6" 

Horizontal direct action high pressure, ----- 3*6 

w=k D a y's. 


Example. Require the weight W=1 of a pair of Horizontal direct ac¬ 
tion propeller engines of D= 72, S^=36 inches, k= 4-6. 

W= 4 , 5X'12 2 |/36 = 139968 lbs. for one cylinder, multiplied by 2=279936 
lbs. the Aveight required. 

For trunk engines must be taken the largest diameter. 

Practical Thickness in Decimals of an Inch of Good 
Plate Iron in Steam-boilers, Single Riveted. 

P = steam pressure in pounds per square inch above atmosphere. 


Press. 


Diameter of Boiler in Inches. 


p. 

10 

15 

30 

25 

30 

35 

40 

50 

60 

70 

80 

90 

100 

130 

150 

300 

10 

.10 

.10 

.11 

.11 

.12 

.12 

.13 

.13 

.14 

.14 

.15 

.15 

.15 

.16 

.17 

.20 

15 

.10 

.10 

.11 

.12 

.13 

.13 

.13 

.14 

.15 

.15 

.16 

.18 

.19 

.19 

.22 

.25 

20 

.11 

.11 

.12 

.12 

.13 

.14 

.14 

.15 

.16 

.17 

.18 

.20 

.20 

.22 

.26 

.30 

25 

.11 

.12 

.12 

.13 

.14 

.15 

.15 

.16 

.18 

.19 

.20 

.22 

.23 

.25 

.30 

.35 

30 

.12 

.13 

.13 

.14 

.14 

.15 

.16 

.18 

.19 

.20 

.22 

.24 

.25 

.28 

.33 

.40 

40 

.12 

.13 

.14 

.15 

.16 

.16 

.18 

.20 

.22 

.24 

.26 

.28 

.30 

.34 

.40 

.50 

50 

.13 

.14 

.15 

.16 

.18 

.18 

.20 

.22 

.25 

.28 

.30 

.33 

.35 

.40 

.47 

.60 

60 

.14 

.14 

.16 

.17 

.19 

.20 

.22 

.25 

.28 

.32 

.34 

.37 

.40 

.46 

.55 

.70 

70 

.14 

.15 

.17 

.18 

.20 

.22 

.24 

.28 

.31 

.35 

.38 

.42 

.45 

.52 

.60 

.80 

80 

.15 

.16 

.18 

.20 

.22 

.23 

.26 

.30 

.34 

.38 

.42 

.46 

.50 

.58 

.70 

.90 

90 

.15 

.17 

.19 

.21 

.24 

.25 

.28 

.32 

.37 

.42 

.46 

.50 

.55 

.60 

.77 

1.0 

100 

.15 

.18 

.20 

.22 

.25 

.27 

.30 

.35 

.40 

.45 

.50 

.55 

.60 

.70 

.85 

1.1 

120 

.16 

.19 

.22 

.25 

.28 

.31 

.34 

.40 

.46 

.52 

.58 

.60 

.70 

.80 

1.0 

1.3 

150 

.17 

.22 

.26 

.30 

.33 

.36 

.40 

.47 

.55 

.60 

.70 

.77 

.85 

1.0 

1.2 

1.6 

200 

.20 

.25 

.30 

.35 

.40 

.45 

.50 

.60 

.70 

.80 

.90 

1.0 

1.1 

1.3 

1.6 

2.1 



















































Punching and Sheering. 


435 


Punching Iron Plates. 

To punch iron plates of from ^ to 1 inch thick requires 24 tons per 
square inch of metal cut; that is, the circumference of the hole multi¬ 
plied by the thickness of the plate is the area cut through. 

Letters denote. 

M = diameter of the punch or hole. 

D = diameter of the hole in the die. 

t = thickness of the iron plate. 

All dimensions in 16ths of an inch. 

W = weight or force in pounds required to punch the hole. 

W= 660 t d. D = d-\-0-2t. 

Example 1. An iron plate of t— 6 sixteenths of an inch thick, and the 
hole to be d— 12 sixteenths in diameter. Required the force W—1 
14^=660X 6X12 =47520 lbs., the answer. 

Example 2. Under the same conditions require the diameter D=1 of 
the die. 

U=12+0-2X6 = 13’2 sixteenths. 

Example 3. Required the diameter of piston for a direction action 
steam punch, for the plate and hole as in example 1, pressure of steam to 
be 50 lbs. per square inch. 

Force 47520=^X60 of which area of piston will be A=—= 950-4 
square inches, which answers to a diameter of 34-8, say 36 inches. 

Shearing Iron Plates. 

It requires the same force per section cut, for shearing as for punching, 
namely ? 20 to 24 tons per square inch. If the shears are good, sharp, and 
well adjusted, 16 tons may be sufficient. 

When the cutters in the shears are inclined to one another, the area 
cut, will be the square of the thickness of the plate multiplied by half 
the cotagent for the angle of the shears. Let -y=angle of the shears, W 
and t same as for punching. 

IF=88 f cot.v. 

Example 4. What force is required to cut a half inch plate t=8 sixteenths 
with a pair of shears forming an angle of v -12 3 '? Cot.l2°=4 , 7. 

JF=88X8‘ 2 X4-7=26470 lbs. 

Atmospheric Columns. 

Wa^er=33-95 feet. 2-3 feet for 1 lbs. per square inch. 

Seawater=33 05 ft. 2‘23 “ “ 

Mercury at 60°=30 inches. 2-05 inches, “ 

Atm. air=28l83 feet. 1912 feet, “ “ 

Atmospheric air Required for each. 

Blacksmith’s forge, - - 100 to 200 

Charcoal forge, - - - - 400 to 500 I 

Finery forge, - - 800 to 1000 >• 

Charcoal furnace, - - 1000 to 3000 

Anthracite furnace - - 2000 to 5000 J 

Cupola. 

In a cupola of 3 feet 4 inches diameter, and 10 feet high, can be melted 
down 1000 lbs. of cast iron, 200 lbs. of bitumninous coal per hour, with a 
blowing machine of 4-5 horses making 1700 cubic feet of air per minute 
into a pressure of 2 - 25 inches of mercury at which the temperature of 
the blast will be about 70° Fah. 


1 


Cubic feet per minute. 













m 


Steam-boiler Explosions. 


Steam-boiler Explosions. 

The steam-boiler is a reservoir of work. Each unit of heat in the steam and 
water is equivalent to a work of 772 footpounds. 

The steam-table gives the units of heat per cubic foot, or per pound, in the 
steam and water at different temperatures and pressures. Work is the product 
of the three simple elements force F, velocity V, and time 1\ or K — F V 1\ 

K 

when the force of the work will be F = - -. When the pressure in any part 

of a steam-boiler is suddenly removed, the entire work in the steam and water 
is at the same time started with a velocity proportionate to the removed pres¬ 
sure. The steam and water, in the form of a foam, strike the sides of the boiler, 
by which the work is suddenly arrested. If the time of arresting the work is 
infinitely small, we see from the above formula that the force of the work will 
be infinitely great, and thus the boiler explodes. 

Steam-boiler explosions are caused in various ways, namely : 

1st. By long use boilers become corroded and, from neglect, give way in some 
unexpected place. 

2d. The general construction with staying and bracing of steam-boilers is 
often very carelessly executed and results in explosion. This kind of explosion 
is often indicated, long before the accident occurs, by leakage of the boiler; 
when the engineer, not suspecting the approaching danger, limits its remedies 
generally to efforts to stop the leak. The leakage from bad calking, or pack¬ 
ing, is easily distinguished from that of bad or insufficient bracing. In the 
latter case the fire ought to be hauled out, the steam blown off, and the boiler 
secured with proper bracing. 

3d. Explosion is sometimes caused from low water in the boiler, but more 
rarely than generally supposed. When the fire-crown and tubes, subjected to a 
strong heat and not covered with water, the steam does not absorb the heat 
fast enough to prevent the iron from becoming so hot that it cannot withstand 
the pressure, but collapses from weakness. 

4th. Steam-boilers often burst by strain in uneven expansion or shrinkage, 
occasioned by the fire being too quickly lighted or extinguished. 

5th. It is a very bad practice to make boiler-ends of cast-iron, composed of a 
flat disc of from two to three inches thick, with a flange of from one to two 
inches thick, with cast rivet-holes. The first shrinkage in the cooling of such a 
plate causes a great strain, which is increased by riveting the boiler to it. Any 
sudden change of temperature, therefore, either in starting or putting out the 
fire, might crack the plate and thus occasion an explosion. Such accident may 
be avoided by making the cast-iron ends concave and of even thickness. 

6th. In cold weather, when the boilers have been at rest for some time, they 
may be frozen full of ice: then, when fire is made in them, some parts are sud¬ 
denly heated and expand, whilst other parts still remain cold, causing an undue 
strain, which may also burst the boilers. Such accident call be avoided by a 
slow and cautious firing. 

7th. Sometimes a great many boilers are joined together by solid connections 
of cast-iron steam-pipes, which expand when heated, whilst the masonry en¬ 
closing the boilers contracts. Should such a steam pipe burst from expansion 
or shrinkage, explosion will likely follow in all the connected boilers, of which 
numerous examples have occurred. Such accident may be avoided by making 
the connection elastic, or free to expand or contract without moving the boilers. 

Steam-boiler explosion is thus not always caused by the pressure of steam 
alone, but often by the expansion and contraction of the materials of the boiler. 
A steam-boiler which is perfectly safe with a working pressure of 200 lbs. may 
explode with a pressure of 20 lbs. to the square inch. 

The bursting of a boiler is a preliminary process to explosion. A boiler may 
burst without exploding. A boiler full of steam may burst, but never explode. 
It is the work in the heated water which makes the explosion. 

The sudden disturbance in the water by the forming of foam generates a 
high heat, and consequently a corresponding high steam-pressure is formed 
over the original pressure, and thus causes a disastrous explosion. 

It is evident from the results of explosions that a much higher pressure had 
been acting than the normal working pressure. 





Destructive Work of Steam-boiler Explosion. 


4 3/ 


Resti-uctive Work of Steam-boiler Explosion. 

When a steam-boiler explosion takes place, the enclosed Mater is resolved 
into one volume of boiling hot water, and one volume of steam, as IoIIom s :— 

Notation of letters prior to explosion. 

W r — weight in pounds of the water under full steam pressure in the boiler. 
w'= pounds of water evaporated in the explosion. 

It = units of heat per pound in the water W. 

H — units of heat per pound in the steam of pressure P. 

H'— units of heat per cubic foot in the steam P 
P — pressure of steam in pounds per sq. in. 

V — volume coefficient of steam. 


Then 


w' — 


W f H (7i — 180-9) 
824'8 


1 


The destructive work K will be in footpounds. 

K = 5^728 H WV {h ~ 180 ' 9 X F— log P- 1-6848298). 

The number vdiich expresses the destructive work of an ordinary steam- 
boiler explosion in footpounds is so large as to be inconceivable to the mind; 
for Mdiicli it is proposed to express explosions by a larger unit, as workmandays, 
of 19S0000 footpounds each, or 2564-75 units of heat. (See page 262.) 

The workmandays of explosion will be 


W' HP{li — 180-9)(F— 1)(23log.P— 1-6848298) 
11341440 IP V 


Example. —In the explosion at CORNELIUS & BAKER’S, Philadelphia, April 
25,1864, the boilers contained about W* = 14750 lbs. of water; the steam- 
pressure P = 80 lbs., H = 1177-05, IP = 223-82, V= 328-08, and h = 282-78. 
Required the workmandays of the explosion. 

14750 X 1177-05 X 80(282-78 —180-9)(328-08 —1)(2-3 X log.80—1-6848298) 

C ~ ~ " 11341440 X ^23-82 X 328-08 _ 


149-63 workmandays; or 150 men in one day could do the work of the explosion. 

The destructiveness of an explosion is thus proportioned to the quantity of 
water and units of heat in the boiler. The steam in a boiler, prior to the ex¬ 
plosion, does very little or no damage in the explosion. The work concentrated 
in a given volume of steam expanded into the atmosphere is 

K = C TP (0-1518 log.P — 0-1771987). 4 

C — cubic feet of steam in the boiler. 

Precaution against Fire on Steamboats. 

Each steamer should have three buckets for every 100 tons measurement, 
plus 10 buckets. That is, a steamer of 800 tons should have 8 X 3 + 10 = 34 
buckets. Also oue axe for every 5 buckets. 


U. S. Steam-boilers Inspector’s Rule for Strength of Boilers. 

Multiply one-sixth (%) of the lowest tensit strength found stamped on any 
plate in the cylindrical shell, by the thickness expressed in parts of an inch of 
the thinnest plate in the same cylindrical shell, and divide the product by the 
radius or half the diameter of the shell expressed in inches, and the quotient 
will be the steam pressure in pounds per square inch allowable in single 
riveted boilers, to which add twenty per centum for double riveting. 

S = breaking.strain in pounds per square inch stamped on the plate. 

t = thickness of the plate in fraction of an inch. 

D = diameter of the boiler in inches. 

P = steam pressure in pounds per square inch. 


P = 


St 

3D 


D = 


St 
3 P 


3 DP 


S = 


3 DP 


S 


t 












433 


Superheating. 


SUPERHEATED STEAM. 

The Author’s experience in superheated steam has been sufficient to 
convince him of its great importance. It appears that in order to utilize 
the maximum effect of steam or at least to attain the maximum quality 
I of expansion, it is not necessary to overheat it after a pure steam is 
formed, that is, when all the small particles and bubbles of water are 
evaporated. Water which accompanies the steam in such a form has 
the same temperature as that due to the surrounding steam pressure, 
prevents it to vaporise; but when it passes through the superheating 
apparatus the temperature is greatly increased while the pressure re¬ 
mains the same because it being in connection with the steamroom in 
the boiler allows the water to vaporise and a pure steam may be formed. 

If steam with particles of water is admitted into the cylinder part of 
the stroke and then allowed to expand, it is generally found that the end 
pressure does not come up. to that due by theory, from which it has been 
pronounced that the expansive quality of steam does not follow that of a 
perfect gas, and that steam has condensed during the stroke ; but if we 
knew the cubic containt of all the particles of water and subtracted that 
from the cubic containt of the steam it might be found that its expansive 
quality is not so far from that of a perfect gas. It appears also that the 
expansive quality is diminished by overheating pure steam. 

The small particles of water contain a great deal more caloric per 
volume than the surrounding steam, consequently when admitted into 
the condenser a good vacuum cannot be formed so well as with pure 
steam. It is therefore of great importance to pay particular attention to 
the superheating of steam, otherwise economy by expansion will not be 
realized to the extent herein given by formulas and tables. It is also of 
great importance for expansion that the piston and steam valves are per¬ 
fectly tight. 

SUPERHEATING APPARATUS. 

The accompanying figure represents a 
superheating apparatus such as the Au¬ 
thor has built it in Russia, and is found 
to answer exceedingly well. The figure 
is a section of the forend of an ordinary 
tubular boiler with steamdrum and up¬ 
take. The chimney is made a great deal 
wider in the steamdrum and contracted 
to the usual size at e, of 0T6 times the 
area of the firegrate; if a strong fan blast 
is app lied it may be better to contract it 
to OTl f—| . In the inside of the chimney 
are placed a number of copper tubes o, 
a, b, b , with flanges screwed to the side ; 
the area of these tubes should be about 
four times that of the steampipe c. In 
the steamdrum is riveted steamtight a 
conical plate d, d, so that the steam can¬ 
not pass to the top without passing the 
superheating pipes. This superheating 
apparatus is in successful operation in 
three first class passenger steamers on 
the River Volga in Russia, each of 600 
actual horses, and one in a steamer of 
100 actual horses on the Black Sea. 

The steamdrum can be placed around 
the chimney separately from the boiler 
and the steam led either above or below 
the plate d, d, by pipes from the steam- 
room, as may suit the circumstances. 

This superheating apparatus may also be well suited for locomotives. 











































Giffard Injector. 


439 

—I 



THE GIFFARD 
INJECTOR, 

Sellers’ Patent. 

The following table has 
been furnished by William 
Sellers & Co., Philadelphia, 
manufacturers of this injec¬ 
tor. It gives the quantity 
of water injected per hour 
in cubic feet. 

The first column No. is 
the size or diameter of the 
throat in French millime¬ 
ters. The last column is the 
size in 16ths of an inch. 



Capacity and Size of Giffai-d’s Injector. 


Size 

Pressure of stee 

im in 

pou 

nds p 

er sq 

uare 

inch 

above 

atmosphere. 

Size 

No. 

10 

30 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

130 

150 

16ths 

2 

8.3 

9 

9.7 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

15.3 

16.0 

16.7 

18.1 

1.26 

3 

19.3 

21.0 

22.8 

24.6 

26.3 

28.1 

29.9 

31.6 

33.4 

35.2 

37.0 

38.7 

40.5 

44.1 

1.89 

4 

36.6 

39.6 

42.7 

45.9 

49.0 

52.1 

55.3 

58.4 

61.6 

64.7 

67.8 

71.0 

74.1 

80.4 

2.5 

5 

57.6 

62.5 

67.4 

72.3 

77.2 

82.2 

87.1 

92.0 

96.9 

102 

107 

112 

116 

126 

3.6 

6 

83.5 

90.6 

97.7 

105 

112 

119 

126 

133 

140 

147 

155 

162 

169 

183 

3.78 

7 

114 

124 

133 

143 

153 

162 

172 

182 

192 

201 

211 

221 

231 

250 

4.41 

8 

149 

162 

174 

187 

200 

213 

226 

239 

251 

264 

277 

290 

303 

328 

5.04 

9 

189 

205 

221 

237 

254 

270. 

286 

302 

318 

334 

351 

367 

383 

415 

5.67 

10 

234 

254 

274 

294 

313 

333 

353 

373 

393 

413 

433 

453 

473 

513 

6.30 

12 

337 

366 

395 

423 

452 

481 

510 

539 

567 

596 

625 

654 

682 

740 

7.56 

14 

451 

491 

531 

571 

611 

651 

691 

731 

771 

811 

851 

891 

931 

1011 

8.80 

16 

600 

651 

703 

784 

805 

857 

908 

959 

1010 

1062 

1113 

1164 

1215 

1318 

10.1 

18 

760 

825 

890 

955 

1020 

1085 

1149 

1214 

1279 

1344 

1409 

1474 

1539 

1669 

11.3 

20 

939 

1019 

1099 

1179 

1259 

1339 

1420 

1500 

1580 

16601 

1740 

1820 

1900 

2061 

12.6 


Method of Working the Injector. 

First. —See that the steam-plug is closed down, and waste-valve stem is raised. 

Second. —Admit steam from boiler to Injector, which should cause the water to 
flow from the waste pipe. 

Third.— Turn up the steam-plug until the waste valve can be closed without 
causing the Injector to cease working. 

Fourth.— Turn up the steam-plug to increase the delivery, and down to decrease 
it. When this Injector has to lift its supply water, the steam valve between the 
Injector and boiler must be opened very slowly, until the water flows out of the 
waste pipe. 

N. B.—A failure to work will always be indicated by an escape of steam and 
water from the waste check attached to check valve in water-supply pipe. 

















































































440 


Blowing Machines. 


BLOWING MACHINES. 

Letters denote. 

*S= strok^ in ^ef 101163 ’} blowing cylinder double acting. 

i = part of the stroke S' under which the air compresses from the 
atmospheric density to that in the reservoir. 

F=mean resistance in pounds per square inch of the air on the 
cylinder piston. 

P = pressure in pounds per square inch of the blast in the reservoir. 

C = cubic feet of air of atmospheric density, delivered from the blow¬ 
ing cylinder to the reservoir per minute. 

H = actual horse power required to work the blowing engine, includ¬ 
ing 13 per cent, for friction. 
d — diameter of blast pipe in inches, 
n = number of revolutions or double stroke per minute. 

A = area of supply valve to the blowing cylinder in square inches, at 
each end of cylinder. 

p = vacuum in pounds per square inch, on the supply side of the cylinder 
piston, which should not exceed 0-1 lbs. 

V = velocity of the blast through the tuyeres in feet per second. 
v — velocity of the air through the supply valve A, in feet per second, 
which should not exceed 100 feet. 
a = area of the orifice or tuyeres in square inches. 
h = height of mercury in inches, in the gauge on the blast reservoir. 

L — length of the blast pipe in feet from the receiver to the tuyeres. 
k — volume coefficient, see Table. 

*=. temperature Fah. of the blast caused by compression or heating. 
Example 1. Formulce 8. For an Anthracite blast furnace is required 
4000 cubic feet of air per minute, under a pressure of 6 inches mercury. 
Required the horse power necessary for the blowing machine 1 The ef¬ 
fectual resistance F=2-365 lbs. see Table. Assume the vacuum to be 
t>=0-09 lbs. 

T _ r . „ 4000 (2-365+0-09) „ . 

We have H =- - -- =49-6 horses. 

198 

Example 2. Formulce 10. Suppose the blast cylinder to be D=144 inches 
diameter with S=15 feet stroke. Required the number of double strokes 
per minute n=1 

96X4000 

144 3 X15 

Example 3. Formulce 9. Under the above conditions, require the area 
of the supply valves A=1 when the velocity v=105 feet, per second. 


n = 


= 12-3 the answer. 


A — 


144 2 X15X12-3 
”40X105' 


= 911 square inches. 


Capacity of Blast Reservoir should not be less than the following 
proportions, but more is better. 

For one single acting cylinder, 204 

For one double acting cylinder, 10 Vtimes the capacity of one cylind’r. 
Two double act. cyl. cranks at 90° 5) 

One double acting cylinder, same as two single acting. The more 
cylinders the less capacity required for blast reservoir. 


F=24 Q\/\ i l+0'00208i), 
P=14-7 (*—1), 
t — 32 


P = 


33-55’ 


*==32+493 (k— 1), 
t = 33-55 P+ 32, 

* 14-7’ 








Blowing Machines. 


441 


Formulas for Blowing Machines. 

Sh 

““30+ h' 

- - 1 

C=l:S3ah(30-\-h),6 

D>Sn(F+p) 
19000 ’ 

s/C- +10 £ 

3 

11 

P= 0-49 A, 

- - 2 

y=350 s/P 

12 

c- ff s * 

_ _ Q 

H C(F + P) 8 

1 

OQ ^ 

II 

13 

“ 96 ’ 


~ 198 

198 H 
F+p ’ 

- - 4 

. Z> S n 

A= - j7r , - - 9 
40v 

D x S* r? 

P ~ 1^80000000 A* 

14 

ah V 
— 26 ’ 

- - 5 

a 

II 

i 

1 

M 

O 

30+ h 

C ~~30 ’ ' 

15 







Table for Blast and Blowing Machines. 


Volume and iicmperat. 

Guage in inches. 

Pressure lbs. sq inch. 

Stroke. 

Velooitv. 

k 

t 

water. 

h 

P 

F 

1 

V ' 

1*002 

33° 

1 

0-073 

0-036 

0-032 

0-0024 

72 

1*005 

34’5 

2 

0-147 

0-079 

0 063 

0-0019 

102 

1-007 

35-5 

3 

0-220 

0-108 

0-095 

0-0073 

125 

1-010 

37 

4 

0-294 

0-144 

0-128 

0-0097 

144 

1-012 

38 

5 

0-368 

0-180 

0159 

0-0121 

161 

1*015 

39-5 

6 

0-441 

0 216 

0-191 

0-0145 

176 

1-020 

42 

8 

0-588 

0-288 

0-253 

0-0192 

204 

1-025" 

44-5 

10 

0-736 

0-360 

0-309 

0-0239 

228 

1-030 

47 

12 

0-884 

0-432 

0-379 

0-0287 

249 

1-035 

49-5 

14 

1-030 

0-503 

0-437 

0-0334 

269 

1-043 

53-5 

17 

1-250 

0-612 

0-531 

0-0400 

297 

1-051 

57-5 

20 

1-470 

0-719 

0-623 

0-0467 

322 

1-062 

63 

24 

1-766 

0-863 

0-745 

0-0556 

352 

1-074 

69 

28 

2-060 

1-008 

0-865 

0-0643 

381 

1-082 

73 

31 

2-281 

1-116 

0-955 

0-0706 

401 

1-091 

77.3 

34 

2-501 

1-223 

1-043 

0-0769 

420 

1-100 

82 

37 

2-720 

1-332 

1-130 

0-0833 

438 

1-109 

86-5 

46 

3-000 

1-470 

1-205 

0-0908 

460 

1-116 

90 

47-5 

3-500 

1-715 

1-431 

0-1045 

496 

1-132 

98 

54-3 

4-000 

1-961 

1-636 

0-1178 

530 

1-165 

114-5 

67-7 

5-000 

2-450 

2-010 

0-1431 

693 

1-200 

132 

81-4 

6-000 

2-941 

2-365 

0-1667 

650 

1-265 

164-5 

108-5 

8-000 

3-925 

3-088 

0-2105 

751 

1-400 

232 

163 

12-00 

5-900 

4-389 

0-2859 

918 

1-500 

282 

203-7 

15-00 

7-375 

6-875 

0-3333 

1077 

1-625 

344-5 

254-6 

18-75 

9-217 

8-831 

0-3846 

1393 

1-750 

407 

305-5 

22-49 

11-06 

10-67 

0-4285 

1590 

1-875 

469-5 

356-4 

26-24 

13-90 

11-64 

0*4666 

1760 

2-000 

1 

t 

532 

407-4 

30-00 

i 

14-75 

12-50 

0-5000 

1955 





















































Fans or Ventilator. 


442 


FAN OR VENTILATOR. 

Fans constructed as the accompanying 
figure have been found by the Author who 
has made several of them, to be the most 
effective. 

The vanes are each one quarter of an 
arithmetical spiral with a pitch twice the 
diameter of the fan, that is, each vane should 
be constructed in an angle of 90° from centre 
to tip. Length of fan to be from i to the 
diameter. Inlet to be half the diameter of 
the fan. Number of vanes to be not more 
than six, and not less than four. Six vanes 
work softer and better, but they give no 
better effect than four. 

_ The housing should be an arithmetical 
spiral with sufficient clearing for the fan at a, and leaving a space at b 
about £ of the diameter. Fans of this construction make no noise. 

Letters denote. 

1 = length 161 "} of fan in inches - 

L= lmwth^n^feet" [ blast pipe, to be as straight as possible. 

a = area in sq. in. tuyeres or outlet. 

C = cubic feet of air delivered per minute. 
h = inches of mercury. 
v — velocity in feet per second through a. 
k = volume coefficient, see Table, page 441. 
n = revolutions of fan per minute. 

H— actual horse power required to drive the fan. 

Formulas for Fans. 



W 


d ri> f~~ d l 
50000000 \y 25 a-\-dl 


1 


4 nr 

28*86 \/ 25 o+dl 




dlhn 

24000* 


24000 H 
din ’ 


2 


3 


n 


24000 H 
dlh ’ 


vak 

C -1 

2-6 ’ 


C= 94 aky h 


7 

8 
9 


V — 244]//i 
A —a y' L 


4 

6 


A = 


C /L 
94 k\/ h ' 


10 


Example 1. A fan of d= 36 inches diameter, 1=12 inches, making n —725 
revolutions per minute, area of tuyere being a=25 square inches. Re¬ 
quired the density of the blast in inches of mercury h=1 


Formula 1. 


h = 


sex^ j 36X12 

60000000 \ / 25X25+36X12 


= 0-242 inches. 


Example 2. Under the same conditions require the cubic containt of air 
delivered per minute, C=1 &=1-01 the nearest in the Table. 

Formula 9. C=94X25X 1‘01 >/0-242 = 1167-7 cubic feet. Required the 

horse power H=1 


Formula 2. 


_ 36X12X 0-242X725 
24000 


= 3-16 horses. 


■f 























iron Furnaces. 


413 


BLAST OR IRON FURNACES. 

It is almost impossible to set up the many variable circumstances con¬ 
nected with the performance of Blast Furnaces, into a table form. The 
datas herein given are deduced to an average from the performances of a 
great many furnaces both in America and Europe. 

The accompanying Tables are so arranged that the numbers in Table 
I., multiplied by the numbers in Table II., gives the corresponding charge 
oi Iron ore, lime stone, coal, and the produce of pig iron in pounds per 
24 hours, with the consumption of air in cubic ffibt per minute. 

Table II. contains the effective capacity of blast furnaces in cubic yards. 

Example. It is required to construct a blowing machine for an Anthra¬ 
cite blast furnace of 12 feet diameter of boshes, height of stack 45 feet, to 
be worked with hot blast. Required the produce of pig iron per 24 hours, 
cubic feet of air per minute and actual horse power of the blowing engine 1 

Produce of pig iron 155 Table I.X123 Table II.=19065 lbs. or 8-5 tons per 
24 hours. 

Consumption of air 20X123=2460 cubic feet per minute. Suppose the 
blast to be blown into the furnace at a pressure of P = 2 94^ lbs., vacuum 
in the supply side in cylinder to be p= 0-07 lbs. we shall have the required 
actual power. 2460 (2-364-0-07) 

For amice. 8, p. 287. H = --- - - -= 30-2 horses. 

’ ^ 198 

Table I. Iron or Blast Furnaces. 


The unit being the capacity 

Charge and produce per 24 hours. 

Air 

of the Furnace in 

Iron 

Pig 

Lime 

Coal. 

per 

cubic yards. 

Ore. 

Iron. 

Stone. 

minute. 



lbs, 

lbs. 

lbs. 

lbs. 

eub. feet. 

Soft charcoal 

f Cold blast, 
(.Warm blast, 

535 

700 

215 

292 

198 

256 

400 

350 

24 

19 

Hard charcoal 

( Cold blast, 
t Warm blast, 

670 

875 

270 

365 

245 

320 

400 

350 

24 

19 

Goke 

fCold blast, 

268 

108 

98 

515 

26 

t Warm blast, 

350 

146 

128 

397 

20 

Bituminous 

( Gold blast, 

252 

101 

92 

515 

24 

1 Warm blast, 

327 

136 

120 

397 

19 

Anthracite 

f Gold blast, 

287 

115 

105 

515 

26 

\ Warm blast. 

373 

155 

137 

597 

20 


Table II. Capacity and Dimensions of Iron Furnaces. 


Diameter of 



Height of stack in feet. 



Bnbesinfi. 

25 

30 

35 

40 

45 

50 

55 

60 

8 

40 

44 

47 

51 

54 

58 

62 

65 

9 

50 

55 

60 

64 

69 

73 

78 

83 

10 

62 

68 

74 

79 

75 

91 

96 

102 

11 

75 

82 

89 

96 

103 

110 

117 

123 

12 

90 

98 

106 

114 

123 

130 

139 

147 

13 

105 

115 

125 

134 

144 

153 

163 

172 

14 

121 

133 

145 

155 

167 

178 

189 

200 

15 

140 

153 

166 

178 

191 

204 

217 

230 

16 

160 

174 

189 

203 

217 

232 

247 

261 

17 

280 

197 

213 

229 

245 

262 

279 

295 

18 

202 

220 

239 

257 

275 

293 

312 

330 


L 






































Ml 


Parabolic Construction Of Ships. 


ON THE PARABOLIC CONSTRUCTION OF SHIPS. 

f ho water-lines, frames, areas of frames and water-lines, and the displacement 
of a vessel, are all parabolas of different orders and power, with the vertex in the 
greatest cross-section or dead flat frame. 

The accompanying tables contain ordinates for different parabolas, with the cor¬ 
responding areas, displacement, centre of gravity, meta-centre, tangent for the 
angles of resistance and inflection. 

The lengths from to the stem and stern, and the draft from the water-line 
to the keel, are each divided into eight equal parts, at which the ordinates are 
calculated. 

The first column in the tables contains the exponent n for the parabolas. The 
higher the exponent is, the fuller will the vessel be. The power q of the parabolas 
makes hollow lines when greater than 1, and the higher the power is, the hollower 
will the lines be. The power of the water-lines should never be less than 1. 

The top-line contains the number of the ordinates, which are counted from the 
stem or stern to JJF, or from the keel to load-water line. 

The half breadth of the beam is-the unit for the ordinates of the load-water 
line, and for the frame jgF. 

It requires some experience in selecting the proper exponents and powers, but 
let us suppose, for example, the exponent n = 2 and the power q= 1, for the for¬ 
ward load-water line of a vessel of length l = 100 feet from ^ to stem, and the 
beam B = 30 feet. Then multiply 15 feet by the ordinates in the table, and the 
products will be the corresponding ordinates for the water-line, the whole area of 
which will be (see column a), a = 100 X 30 X 0.6666 = 1999.99, or 2000 square feet. 

The centre of gravity of that area will be (see column e )— 
e = 100 X 0.3750 = 37.5 feet from ?g\ 

Now let us select the exponent n = 5 and the power q = 1 for the frame jg", 
and let d — 12 feet depth from the water-line to keel. Then multiply 15 by the 
ordinates of that exponent and power, and the products will be the corresponding 
ordinates for the frame jgT. The area of the frame Tg' will be (see column a), 
= 30 X 12 X 0.8333 = 299.988, or 300 square feet. 

The depth of the centre of gravity of jgF will be (see column e)— 
e = 12 X 0.4286 = 5.14 feet under the load-water line. 

Let us select the exponent n = 2.25, and power q — 1 for the forward displace¬ 
ment. Then multiply = 300 square feet by the ordinates for that exponeut 
and power, and the products will be the corresponding areas at each frame. The 
forward displacement will then be (see column a)— 

D = 300 X 100 X 0.6923 = 20769 cubic feet. 

The distance from ?£F to the centre of gravity of the forward displacement will 
be (see column e), e = 100X 0.3823 = 3823 feet. 

The operation is precisely the same for the after-body of the vessel, only that 
fuller exponents are generally selected. 

After the lines are laid out as described, the divisions of the frames are made as 
required in the building of the vessel. 

For simplicity in our illustration, let us suppose that the exponents and powers 
are respectively the same for the after-body of our vessel, and 100 feet from $£ to 
stern ; then the displacement will be D — 20769X 2 = 41538 cubic feet, the whole 
length L — 100 X 2 = 200 feet. 

The height of the meta-centrum will be— 


m 


/ _ 


LlPrn 200 x 15 3 x 0.3021 


D 


41358 


= 4.9305 feet. 


The coefficient m = 0.3021 is taken for the exponent and power of the load-water 
line in column m. 

The tangent for the mean angle of resistance to the vessel in motion in water 
will be— 

'Mt 300x1.447 . 

tang. v = ——- =-- = 0.18087 = 

2 V d 2x100x12 

tang. 10° 15', the mean angle of resistance. 








445 


Parabolic Construction of Ships. 


The coefficient t = 1.447 is taken for the exponent and power of the displace¬ 
ment in column t. 

The whole area of the load-water line is a = 2000 X 2 = 4000 square feet 
The depth of the centre of gravity of the whole displacement will be— 


e = 


2 


d 12 


(2 D \ 

0 (9 

41358 \ 

\ a d) 

^ 1 z, 

4000xl2j 


= 5.28 feet, under the wa- 
[ter-line. 


To Calculate tlie Ordinates for tlie Frames. 

Having given the half area of the frame, the ordinate of the load-water line and 
the depth d, multiply the ordinate by the depth, and divide the given area by the 
product, which will give a decimal fraction. Find this fraction in the columns 
a, or area in the tables for frames, multiply the ordinates of that area by the 
given ordinate for the load-water line, and the products will be the corresponding 
ordinates for the frame, 

Example. The given area = 64.5 square feet, the ordinates of the water-line = 

6 feet, and the depth = 18 feet. Then —?^—= 0.599. Find this area in table 

18X6 

q = 1 fop frames, and it will be found to correspond nearly with the ordinates of 
exponent n = 1.5, which will give a full line; but if it is desired that the frame 
should be a reverse curve, then select the area of higher power—for instance, in 
table q = 1.25, the area 0.599 corresponds nearly with the ordinates of the expo¬ 
nent n = 1.75, in which case the frame will reverse at 18 X 0.77 = 13.86 feet from 
the water-line. 


To lay down tlie Ordinates of all tiie Water-lines 

and frames on the drawing direct from the tables without calculation. 

Construct a scale or diagram like that on Plate VII., divided into 100 equal parts 
each way. 

Set off half the beam b of the same scale as the drawing, from B toward C, say 
at a. join a with A, then the line a A , measured from the base A B, is the scale for 
all the ordinates in the load-water line and greatest cross-section jgf. Let us se¬ 
lect the exponent n — 2.5 and power q = 1.5 for the forward load-water line, which 
will be slightly hollow and reverse at 0.782 X l' from ]£[. Measure off the ordi¬ 
nates from the corresponding number on the base A B to the line a A, and set 
them down on the drawing. 

Set off the ordinates from B toward a. and join them with A ; then each line 
forms a scale for the ordinates of the corresponding frames. The fourth ordinate 
V.7469 will be the linear 7i, and the incline line 4 is the scale for the fourth frame. 

Turn the scale round, and set off the half beam b from D toward A; join it with 
C, and proceed in the same way for the after-body of the ship, only select higher 
exponent, for the aft load-water line. 

Scales or diagrams like that on Plate VII., but of larger size, have been printed, 
and can be had in Philadelphia. 


To find tlie Exponents and Powers 

from the lines of a vessel constructed without regard to the parabolic method. 

Divide the lengths from to the stem and stern, and the depth from the load- 
water line to the keel, each into eight equal parts, as before stated. Suppose half 
the beam of the vessel is b = 20 feet. Measure the 2nd and 4th ordinates in the 
load-water line, say 6.68 feet and 15.31 feet, respectively. Divide those ordinates 
by the half beam 20 feet, which will give 0.334 and 0.765. Find these ordinates in 
the tables, which will be found to correspond with exponent n = 3 and power q = 2. 
The area of the water-line will be, supposing the length to be 130 feet from ST 
to stern, 130 X 40 X 0 6428 = 3342.56 square feet. 

All the other exponents and powers are found in the same way. For the dis¬ 
placement, divide the ordinate areas of the frames by Jff and find the correspond¬ 
ing ordinates in the tables, and thus all the properties of the displacement a e 
found by the parabolic method. 
















416 


Parabolic Construction of Ships. 


T .n.q. \ i / L.B.d. \ y ( W.n.q. 
).n.q.) \M n.q.) \ D.n.q. 


Recording Formula 

The form of any vessel may be recorded by one general formula, as follows— 

W. 

D.n.q.) n.q. 

W71,0« ") 

The first part, ——— h l, represents the form of the after-body of the displace- 
^ ’ D.n.q. ) 

ment. 

W.n.q. = exponent and power of the load-water line, D.n.q. = exponent of the 
displacement, and 1= length from $£ to stern-post. 

The middle part, ( L - Bj - ) represents the length, beam and depth of the dis- 

placement, ffi.n.q. tire exponent and power of the frame ,g£. 
r w.Ti.Q. 

The last part, l' \ —represents the fore-body of the displacement. 

^ ( D.n.q. 

The recording formula is not intended for arithmetical operation. 

Recording Formula of tlie Ironclad Dictator.” 

TP 5.5 X 


D4.75X 


1.75] / 240X42 X16 \ m [ >72.75X1-5 

2.25J VST 2.75X0.5/ \ D 3 X 2 


From these data a similar vessel to the hull of the “ Dictator” can be constructed 
by any one familiar with the parabolic method. 

Sailing-Yaclxt. _ 

A well-proportioned sailing-yacht may be set up as follows: 


>73x2 
D 2x2 


/ V ffi 3x4 / l 


W 2.75x2 
D 2x2 


Tlie Formula for tlie Steam-Propeller on Plate X. is— 


>72.5x1 


D 2x2 i 


} 65.625 ( ^ X 3 ° o X ,- -) 84.375 { 
j \ ffl 3 x 2.5 / l 


>72x1.5 
D 2x2 


Angles of tlie Lines. 

The angle of entrance in the stem, or delivery in the stern, of any water-line, is— 

b n 


tan.fl = 


lq 


The angle of the line, at any distance y from ^T, is— 

bnqy n ~ x 


tan.-y: 


l n 


Mr 


The mean angle of resistance or delivery of the displacement is— 

dy. 


tan.v = q 2 b n 2 J 1^1 — y ^ 


2g — 2 yin — 2 

Id l 2n 

The column V in the tables is calculated from this formula. 
The distance y from to the point of inflection, is— 


y 


\nq 


1 




























Parabolic Construction of Ships. 


447 


Explanation of Tables. 

Table I. gives the greatest buoyancy and stability of the displacement with the 
least proportionate resistance. 

Tables II. to IX., inclusive, give hollow lines of which those with high power 
and low exponents are too hollow for the load-water line, but the high exponents 
with high power will suit for the aft load-water line. 

Tables X. to XIII. should not be used for water-lines, but for frames. 

Tables XIV. to XIX. are for frames exclusively. 

Table XX, for elliptic stern and sheer of vessels. 

Table XXI, contains the coefficient of displacement, or the fraction it occupies 
in the parallelopipedon, length, breadth and depth. For light draft and high speed 
the coefficient should be selected toward the corner .490, and for freight and light 
draft toward the corner .797. The first column $%£ n contains the exponent for 
and the tope-line that for the displacement, supposing q — 1 in both cases. 

Table XXII. The first column C contains the coefficient of the vessel, for which 
the tabular number, multiplied by the cube root of the displacement in tons, will 
give the length of the vessel in feet. 

Explanation of tiie Plates VII., VIII., IX. and X. 

Plate VII. is a scale for laying down the parabolic lines directly from the tables 
without calculation, as before described. 

Plate VIII. illustrates the sharpness of water-lines and cross-sections. The num¬ 
bers on the figures correspond with the exponent n in the first column of the 
tables. 

Plate IX. Fig. 1 illustrates the sharpness of full lines from Table I. Fig. 2 are 
cross-sections from the same Table I. Fig. 3 is a scale of displacement and draft 
of water, referred to Table I., in which the exponent n (or ratio r) is taken for the 
areas of the water-lines. This scale also corresponds with the Formulas 6 to 9, for 
finding the displacement t> and draft d. 

Fig. 4 is a scale for laying out the frames. 

Fig. 5 is the spring or rise of beams, for which the ordinates should always be 
taken from the exponent 2, Table 1. 

Plate X. illustrates a propeller steamer constructed on the parabolic method. 

Steamship Performance. 

For similar proportioned vessels the resistance is a function of jgf M 2 , and the 
horse power a function of ]%£ M 3 . The displacement of a vessel is a function of the 
cube of any linear dimension of the same, and the greatest immersed section ]%£ 
a function of the square of any linear dimension of the disp lace ment ; consequently 

the ^ T is a function of any linear dimension, and (]^JF ) 2 = T 2 is a function of 

• therefore, th e re sistance is a function of M 2 ^ T 2 , and the horse power a 

function of MT 2 . 

The tables of steamship performance are calculated by this last formula, in which 
it is assumed that the displacement of the vessel is about one-half of the parallelo¬ 
pipedon, and the length about eight times the beam. 


M 


3 I' 228 H 

\ f 


11 = 


M^T 2 


228 


//228m 3 


On the Form Waves, or the Wave-line. 

v — velocity of the wave in feet per second. 

I — length of wave in feet from top to top. 
h — total height in feet of the wave from top to bottom. 


I' = 1.82j//, Z = 3.14/1, h — 0.3183Z. 


The accompanying table shows the ordinates 
for the w r ave line, the abscissa or base of the line 
from the lowest part to the highest, or half the 
length of the wave being divided into 16 equal 
parts. The ordinates are expressed in parts of 
the height h. 


Ab. 

Ordinates. 

Ab. 

1 

.009607 

9 

2 

.03806 

10 

3 

.0S426 

11 

4 

.14644 

12 

5 

.22221 

13 

6 

.30866 

14 

7 

.40245 

15 

8 

.50000 

16 


Ordinates. 

.59755 

.69134 

.77778 

.85355 

.91573 

.96194 

.99039 

1.0000 















448 


Parabolic Construction of Ships. 


1 


Notation of Letters for tlie Formulas. 

All dimensions are in feet, square feet or cubic feet. 


The Vessel. 

A = area of the immersed hull, 
a = area to load water-line. 
a — area of water-line at the draft S. 

B = breadth of beam. 
h ■ half breadth of beam B. 
d — draft of water, omitting depth of 
keel. 

5 = any draft less than d. 

= difference of draft of water, which 
should not be taken more than 
0.25 feet for the largest, and only 
0.1 for vessels of 100 tons. 

b — differential. 

L = length of vessel in load-water line. 

I = length from fff to stern-post. 

V = length from ]%£ to stem. 

7Q- = area of greatest immersed section. 
D = displacement, cubic feet. 

D = displacement forward of 
q = displacement aft of 
O = displacement at the draft 8. 

T — displacement in tons. 

b A= differential of displacement. 

Centres of Gravity. 

S — distance of centre of gravity of dis¬ 
placement from the stern-post, 
s = distance of centre of gravity of Q 
from 

s' — distance of centre of gravity of D 
from 

e = centre of gravity in the tables. 
e! = depth of centre of gravity of dis¬ 
placement under the load-water 
line. 

Parabolas. 

n and q — exponent and power. 

(3 = any ordinate at distance y from < £Q\ 
r = exponent of the displacement from 
load-water line to keel, supposing 
9 = 1 . 


Stability of the Vessel, 
m = meta-centre for load-water line in 
the tables. 

m! = height of meta-centre above centre 
of gravity of displacement. 

P = the real stability of a ship. 

Q = momentum of stability in foot-tons. 
g = height between centres of gravity 
of vessel and displacement when 
in equilibrium. 

o — horizontal distance between the two 
centres of gravity when out of 
equilibrium. 

o' — horizontal distance between centre 
of gravity of displacement when 
in and out of equilibrium. 
z = careen-angle of the vessel. 

C — coefficient of displacement, Table 
XXI. 

Performance. 

H— horse-power required to move the 
vessel. 

M = speed in knots per hour. 

R = resistance to the vessel in pounds. 
t — tangent in the tables. 
v = mean angle of resistance to the ves¬ 
sel moving in water. 

V = mean angle of delivery of the aft 
displacement. 

7c — coefficient of friction in pounds per 
square foot of the immersed sur¬ 
face of the vessel. 

Surface. Coefficient, Jc. 

For polished metallic surface, . 0.003 
Ordinary copper sheeting, . . . 0.005 
Wood, smooth-planed, .... 0.007 
Rough castings of iron or brass, . 0.009 
Fouled with barnacles and sea¬ 
weeds, .0.015 

Fouled with barnacles and oyster- 
shells, . 0.020 


^ = [a+2(® + d_L)] J_ D _. ' 

* i L d 

Explanation, of tlie Fonnalas, witli Examples. 

The whole method of the Parabolic Construction of Ships is based upon the 
Formula 1. The formula looks very simple, but when developed into its combina¬ 
tions with the form of a ship, it becomes a very complicated affair, which requires 
a separate treatise on ship-building for proper explanation. 

Example for Formula 5. The length of a vessel is L = 250 feet, of Avhich l = 100 
feet from $£ to stern-post, and V = 150 feet from to stern. The centres of 
gravity of the fore and after body of the displacement have been found as before 
described, to be s = 48 feet aft of and s’ = 68 feet fore of m. Required the 
centre of gravity of the whole displacement from the stern-post: 


Formula 5. 


S = 


100(10-0 —68)+ 150 (100 + 48) 


250 


= 116 feet. 









Pa rabolic Construction of Ships. 


Ml 


Formulas for the Parabolic Construction of Ships, 

Stability and Performance. 


• • • 

Centres of Gravity. 

D 




SLd — D 
d v 

2r+l' 

d 


l( 2 -—) 

V a d! 


S= 


l(l-s')+l'(l + 8) 


Displacement between small 


dif. of Draft. 


. r+1 /od(r+l) r 

— 


6 


btf = 


b o r id 


a 




bt> = a bcby^ 


r /4 

a ~ a \d 


9 


Stability in the careen of 
vessels. 

b 3 

m' = — (mV + ml). 10 
IF ‘ 

Q=T sin.z(m/ =t .11 

o = sin. 2 (m / 12 


o / = m / sin.z. 


m / =t g 


vv 


Q = To. 


Resistance to and Performance of Vessels in Motion. 

i2 = 2.858iir 2 r^(0.9/ZK^+0.1 T /im7F) + ^"l. . 
L ^L -1 

22450H 


ST t 

tan.p = — . 

Vd 

. 17 

H- 

RM 

22450' * ’ 

19 

Wt 

tan. V = -—• . . 
Id 

. 18 

M- 

22450JT 

R 

20 


i2 = - 


M 


Friction = 


Ah 

1 Yf 


13 

14 

15 

16 

21 

22 


Example for Formula 8. A vessel of D = 150000 cubic feet of displacement, 
a = 10000 square feet area of load-water line at d — 20 feet draft. Required, the 
exponent r? and how much b> can the vessel be loaded per inch of difference of 
draft at S = 15 feet draft ? 

T. 7 a 15000 _ . 

Formula 2. r = ————— _ = 3, the exponent. 


40000x20-150000 


Formula 8. 


b> = 


10000x1 3 115 


= 728 cubic feet. 


12 V 20 

which in salt water will be 728 : 35 = 20.8 tons per inch of draft at $ = 15 feet. 

Example for Formula 10. A vessel of D — 150000 cubic feet, half beam 6 = 20 
feet, V = 170 feet, n — 3.5, q = 2, for which meta-centre in the table is m= 0.350, 
ind / -= 180, n = 2.5, q — 2, for which m = 0.277. Required, the height of the 
meta-centre above the cent. gr. of the displacement ? 

20 3 


Formula 10. m / 


■ (0.350 x 170+0.277 x 180) = 5.832 feet. 


150000 

The momentum of stability will be, when T = 150000 : 35 = 4288.3 tons, careen 
angle 21 == 10°, and the height between the two centres of gravity g .== — 1 foot, or 
tlie cent. gr. of the vessel is one foot above the cent. gr. of displacement. 

Formula 11. Q = 4288.8 sin.lO 0 (5.832 — 1) =3591.9 foot-tons. 

Divide the momentum of stability by the length of the lever upon which the 
force is applied to careen the vessel, and the quotient will be the force required iu 
tons. 


29 































4.30 Parabolic Construction of ships. 


1 


Table I. 


Power q = 1. 

Pull lilies. 


Kxp. 



Ordinates. 



Prod. 

0. &T 

Meta. 

lies. 

Intiec- 

n 

1 

2 

3 

4 

5 

6 

7 

a ffil> 

e 

m 

t 

tion. 

o 

.2344 

.4375 

.6-94 

.7500 

.8591 

.9375 

.9814 

.6666 

.3760 

.3021 

1.333 


- 125 

.2470 

.457 4 

.6317 

.7707 

8756 

.9474 

.9879 

.6300 

.3788 

.3162 

1.392 

! 

2.2ft 

.2595 

.4765 

.6527 

.7898 

.8899 

.9588 

.9907 

.6923 

.3823 

.3288 

1.447 

0 

2.575 

.2716 

.4950 

.6725 

.8072 

.9026 

.9628 

.9928 

.7017 

.3857 

.33*4: 

1.50C 

1 

2.5 

.2838 

.5129 

.6912 

.8232 

.9149 

.9687 

.9945 

.7142 

.3888 

.3502 

1.56l 


2.625 

.2954 

.5300 

.708S 

.8378 

.9238 

.9737 

.9957 

.7241 

.3919 

.3612 

1.622 

a 

2.75 

.3073 

.5466 

.7254 

.8513 

.9326 

.9779 

.9967 

.7333 

.394S 

.3688 

1.682 

<a> 

2.875 

.3185 

.5627 

.7411 

.8637 

.9404 

.9814 

.9975 

.7419 

.3974 

.3776 

1.738 

a 

rC 

**. 

.3390 

.5781 

.7558 

.8750 

.9476 

.9841 

.9980 

.7500 

.4000 

.3857 

1.800 

§ 

3.25 

.3521 

.6074 

.7829 

.8949 

.9587 

.9889 

.99S8 

.7647 

.4047 

.4010 

1.922 

1 

3.5 

.3733 

.6346 

.8070 

.9116 

.9677 

.9922 

.9993 

.7777 

.4091 

.4144 

2.04( 

1 

3.75 

.3939 

.6600 

.8284 

.9256 

.9747 

.9945 

.9996 

.7894 

.4130 

.4263 

1 2.166 

O 

4. 

.4138 

.6836 

.8474 

.9375 

.9802 

.9961 

.9997 

.8000 

.4166 

.4376 

j 2.285 

<v 

w 

4.5 

.4517 

.7260 

.8794 

.9559 

.9879 

.9980 

.9998 

.8181 

.4230 

.4570 

2.534 


5. 

.4871 

.7627 

.9016 

.9687 

.9926 

9390 

.9999 

.8333 

.4286 

.4731 

2.775 

u 

6. 

.5512 

.8220 

.9464 

.9814 

.9972 

.9997 

.9999 

.8571 

.4375 

.5000 

3.270 

'ot 

8. 

.6564 

.8999 

.9767 

.9961 

.9 )96 

.9999 

.9999 

.3888 

.4500 

.5351 

4.272 

<X> 

a 

10. 

.7369 

9437 

.9909 

.9990 

.9998 

.9999 

1.000 

.9090 

.4583 

.5585 

5.265 


12. 

.7986 

.9683 

.9964 

.9997 

.9939 

1.000 

1.000 

.9231 

.4643 

.5768 

6.275 

rS 

16. 

8819 

.9900 

.9991 

.9998 

1.000 

1.000 

1.000 

.9412 

.4720 

.6060 

8.273 



Table II. 

Power <i = 

1.35. 


Hollow lines. 


2. 

.1631 

.3558 

.5384 

.6)79 

.8-74 

.9225 

.9805 

.6243 

.3528 

.2778 

1.288 

.8165 

2.125 

.1741 

.3761 

.5631 

.7222 

.8470 

.9347 

.9850 

.6399 

.3572 

.2911 

1.333 

>335 

2.25 

.1852 

.3959 

.5866 

.7445 

>644 

.9451 

.9884 

.6538 

.3623 

.3010 

1.380 

.8478 

2.375 

.1961 

.4152 

.6090 

.7651 

.879 S 

.9538 

.9910 

.6666 

.3660 

.3154 

1.428 

.8510 

2.5 

.2072 

.4340 

.6302 

.7841 

.8948 

.9611 

.9931 

.6788 

.3700 

.3262 

1.476 

>728 

2.6 5 

.2178 

.4523 

.6504 

.8006 

.9057 

.9673 

.9917 

.6900 

.3739 

.3462 

1.521 

.8832 

2.75 

.2288 

.4700 

.6695 

.8178 

.9165 

.9725 

.9959 

.7000 

.3777 

.3558 

1.573 

.8922 

2>75 

.2393 

.4837 

.6876 

.8326 

.9260 

.9708 

.9968 

.7094 

.3814 

.3548 

1.622 

.8993 

3. 

.2502 

.5041 

.7048 

.84 3 

.9350 

.9.805 

.9976 

.7137 

.3850 

.3632 

1.670 

.9055 

3.25 

.2712 

.5362 

.7365 

.8704 

.9487 

.9862 

.9985 

.7338 

.3904 

.3794 

1.7 65 

.9148 

3.5 

.2918 

.5665 

.7649 

.8908 

.9598 

.9902 

.9991 

.7485 

.3950 

.3945 

1.863 

.9212 

3.75 

.3121 

.5949 

.7903 

.9080 

.9685 

.9931 

.9995 

.7616 

.4010 

.4065 

1.975 

.9260 

4. 

.3319 

.6216 

.8130 

.9225 

.9753 

.9951 

.9997 

.7733 

.4010 

.4183 

2.092 

.9306 

4.5 

.3703 

.6701 

.8516 

.9452 

.9'49 

.9975 

.9999 

.7940 

.4111 

.4390 

2.29S 

.9384 

5. 

.4069 

.7127 

.8823 

.9611 

.9907 

.9988 

.9399 

.8108 

.4183 

.4563 

2.492 

.9450 

6. 

.4749 

.7827 

.9260 

.9805 

.9965 

.9997 

1.000 

.8379 

.4283 

.4845 

2.873 

.9555 

8. 

.5908 

.87 65 

.9710 

.9951 

.9995 

.9999 

1.000 

.8733 

.4.397 

.5235 

3.635 

.9694 

10. 

.6828 

.9274 

.9886 

.9988 

.9999 

1.000 

1.000 

.8952 

.4502 

.5182 

4.515 

.9787 

12. 

.7550 

.9607 

.9950 

.9993 

1.000 

1.000 

1.000 

.9120 

.4593 

.5676 

5.150 

.9815 

IG. 

. s ' 547 

.9875 

.9994 

.9999 

1.000 

1.000 

1.000 

.9323 

.4684 

.59651 

7.225 

.9858 


Table III. 

Power q = 

1.5. 

Hollow lines. 


2. 

.1135 

.2891 

.4757 

.6495 

.7966 

.9077 

.9766 

.5897 

.3395 

.2576 

1.250 

.7071 

2.125 

.1227 

.3093 

.5020 

.6766 

.8193 

.9222 

.9820 

.6057 

.3413 

.2706 

1.291 

.7285 

2.25 

.1322 

.3289 

.5273 

.7019 

.8396 

.9344 

.9861 

.6205 

.3489 

.2834 

1.330 

.7484 

2 375 

.1416 

.3483 

.5515 

.7253 

.8576 

.9448 

.9893 

.6347 

.35:14 

.2953 

1.370 

.7666 

2.5 

.1512 

.3873 

.5746 

.7469 

.8751 

.9535 

.9917 

.6476 

.357 6 

.3065 

1.410 

.7820 

2.625 

. 1606 

.3771 

.5967 

.7669 

.8879 

.9608 

.9936 

.6595 

.3619 

.3’70 

1.450 

.7958 

2.75 

.17D4 

.4042 

.6178 

.7855 

.9006 

.9670 

.9951 

.6702 

.3654 

.3266 1 

1.490 

>080 

2.875 

.1797 

.4221 

.6380 

.8027 

.91)9 

.9722 

.9962 

.6802 

.3689 

.3362 

1.532 

.8192 

3. 

.1896 

.4396 

.6571 

.8185 

.9225 

.97 66 

.9971 

.6' 93 

.3723 

.3449! 

1.576 

.8300 

3.25 

.2099 

.4734 

.6928 

.8465 

.9387 

.9831 

.9982 

.7068 

.3781 

.36151 

1.663 

.8473 

3.5 

.2281 

.5056 

.7249 

.8704 

.9519 

.9883. 

.9390 

.7231 

.3836 

.3760 

1.753 

.8612 

>.75 

.2172 

.5302 

.7540 

.8906 

.9623 

.9917 

.99)4 

.7372 

.3888 

.38971 

1.853 

.8732 

4. 

.2662 

.5652. 

.7801 

.9077 

.9705 

.9941 

.9996 

.7500 

.3932 

.4021 

1.952 

.8838 

4.5 

.3035 

.6186 

.8246 

.9346 

.9819 

.9971 

.99)8 

.7723 

.4018 

.4238 

2.128 

.8980 

5. 

.3400 

.6661 1 

.8601 

.9535 

.98S9 

.9385 

.999 ) 

.7909 

.4092 

-4420 

2.300 

.9072 

6. 

.4092 

.7453 ' 

.9119 

.9767 

.9:58 

.9996 

1.000 

.8202 

.4202 

.4719 

2.644 

.9247 

8. 

5318 

.8536 { 

.9651 

.9941 

.9991 

.9999 

1.000 

.8593 

.4356 

.5132 

3.344 

.9451 

1 '. 

.6326 

.9115 

.9864 

.9985 

.9999 

1.000 

1.000 

.8836 

.4461 

.5391 

4.115 

.9585 

12. 

.7136! 

.9530 

.9947 

.9997 

.9999 

1.000 

1.000 

.9024 

4547 

.5598 

4.775 

.9666 

16. 

.82S2| 

.9850 j 

.9992 

.9999 

1.000, 

1.000 

1.000 

.9250 

4619 

.5887 

6.253 

.9735 
































































































Parabolic Construction of Ships. 4^1 



Tn ble I\ 

r 

Power q = 

1.75. 


Hollow lines. 

— 

Kxp. I 



Ordinates. 



Prod. 

0. gr. 

Meta. 

Res. 

Inflec- 

n 

1 

a 

3 

* 

5 

6 

7 


« 

m 

t 

tion. 

2 - I 

.078.) 

.2353 

.4203 

.6044 

.7670 

.8932 

.9728 

.5592 

.3252 

.2412 

1.228 

.6345 

2.125 

.0865 

.2544 

.4476 

.6340 

.7928 

.9098 

.9790 

.5761 

.3308 

.2538 

1.260 

.6600 

2.25 

.0941 

.2733 

. 1739 

.6617 

.8454 

.9240 

.9838 

.5919 

.3358 

.2665 

1.291 

.6822 

2.825 

.1022 

.2921 

.4994 

.6874 

.8359 

.9359 

.9875 

.6065 

.3400 

.2786 

1.324 

.7018 

2.5 

.1104 

.3108 

.5240 

.7415 

.8559 

.9460 

.9904 

.6200 

.3446 

.2900 

1.363 

.7200 

2.625 

.1181 

.3218 

.5 475 

.7337 

.8705 

.9545 

.9925 

.6320 

.34S7 

.3000 

1.400 

.7364 

2.75 

.1269 

.3476 

.5702 

.7545 

.8851 

.9617 

.9943 

.6411 

.3529 

.3105 

1.438 

.7514 

2.875 

.1350 

.3656 

.5919 

.7738 

.8989 

.9677 

.9956 

.6542 

.3568 

.3200 

1.475 

.7650 

3. 

.1437 

.3833 

.6428 

.7916 

.9102 

.9728 

.9966 | 

.6646 

.3604 

.32 )5 

1.511 

.7777 

3.25 

.1609 

.4179 

.6517 

.8234 

.9289 

.9807 

.9980 

.6833 

.3670 

.3464 

1.591 

.7987 

3.o 

.1825 

.4513 

.6871 

.8505 

.9442 

.9864 

.9988 

.7000 

.3736 

.3618 

1.591 

.8175 

3.75 

.1959 

.1833 

.7193 

.8736 

.9562 

.9904 

.9993 

.7158 

.3782 

.3758 

1.673 

.8319 

4. 

.2185 

.5139 

.7484 

.8932 

.9657 

.9932 

.9996 

.7293 

.3834 

.3883 

1.850 

.8430 

4.5 

.2483 

.5710 

. < 985 

.9241 

.9789 

.9)66 

.99:* 

.7521 

.3922 

.4108 

2.010 

.8613 

5. 

.2S40 

.6225 

.8391 

.9460 

.9870 

.9983 

.9999 

.7732 

.4006 

.4400 

2.171 

.8754 

G. 

.3526 

.7096 

.8980 

.9728 

.9951 

.9996 

1.000 

.8050 

.4129 

.4640 

2.500 

.8989 

8. 

.4787 

.8314 

9596 

.9932 

.9993 

.9999 

1.000 

>475 

.4296 

.5041 

3.175 

.9258 

10. 

.5861 

.9009 

.9841 

.9983 

.9998 

1.000 

1.000 

.8734 

.4420 

.5343 

3.878 

.9431 

12. 

.6746 

.9151 

.9938 

.9996 

.9999 

1.000 

1.000 

.8938 

.4503 

.5528 

4.550 

.9433 

16. 

.8026 

.9825 

.9990 

.9999 

1.000 

1.000 

1.000 I 

.9185 

.4615 

5821 

6.059 

.9643 


Table V. 

Power q — 

a. 

Hollow lines. 


2. 

.0549 

.1914 

.3713 

.5625 

.7385 

.8789 

.9690 

.5333 

.8125 

.2273 

1.219 

.5773 

2.125 

.0610 

.2092 

.3990 

.5940 

.7667 

.8976 

.9761 

.5504 

.3160 

.2415 

1.242 

.6033 

3.25 

.0673 

.2271 

.4260 

.6237 

.7920 

.9136 

.9815 

.5664 

.3189 

.2541 

1.273 

.6292 

2.375 

.0738 

.2450 

.4523 

.6516 

8147 

.9271 

.9857 

.5813 

.3278 

.2662 

1.301 

.6517 

2.5 

.0806 

2630 

.4777 

.6777 

.8371 

.9385 

.9890 

.5952 

.3333 

.2770 

1.356 

.6723 

2.G25 

.0873 

.2'10 

.5024 

.7020 

.8534 

.9181 

.9915 

.6083 

.3388 

.2875 

1.666 

.6904 

2.7 5. 

.0914 

.-988 

.5262 

.7248 

.8698 

.9563 

.9934 

.6205 

.3118 

.2979 

1.402 

.7066 

2.875 

.1014 

.3166 

.5492 

.7460 

.8-843 

.9632 

.9934 

.6320 

.3460 

.3974 

1.436 

.722*2 

3. 

.1090 

.3342 

.5713 

.7656 

.8989 

.9690 

.9961 

.6428 

.35 )0 

.3164 

1.472 

.7 -6S 

3.25 

.1240 

.3689 

.6130 

.8008 

.9192 

.9779 

.9977 

.6627 

.3571 

.3255 

1.545 

.79°8 

3.5 

.1394 

.4028 

.6512 

.8310 

.9365 

9841 

.9986 

.6805 

.3636 

.3500 

1.620 

.8175 

3.75 

.1555 

.4356 

.6862 

.8569 

.9501 

.9800 

9992 

.6966 

.3645 

.8638 

1.697 

.8319 

4. 

.1712 

.4673 

.7181 

.8789 

.9608 

.9922 

.9495 

.7111 

.3750 

.3765 

1.773 

.8013 

4.5 

.2045 

.5270 

.7733 

.9137 

.9759 

.9961 

.9998 

.7363 

.38 46 

.4000 

1.929 

.8315 

5. 

.2373 

.5804 

.8184 

.9385 

.9852 

.9980 

.9999 

.7576 

.3929 

.4196 

2.088 

.8483 

G. 

.3038 

.6729 

.8843 

.9690 

.9945 

.9995 

9949 

.7912 

.4U62 

.4547 

2.407 

.8767 

8. 

.4308 

.8098 

.9540 

.9922 

.9992 

.9990 

1.000 

.8366 

.4250 

.4962 

3.060 

.9088 

10. 

.5431 

.8905 

.9819 

.9980 

.9999 

1.000 

1.000 

.8643 

. 4: 75 

.5249 

3.643 

.1)279 

12. 

.6377 

.9379 

.9929 

.9995 

.9999 

1.000 

1.000 

.8861 

.1464 

.5463 

4.390 

.9418 

1C. 

.7778 

.9800 

.9989 

.9999 

1.000 

1.000 

1.000 

.9126 

,4629 

.5763 

5.715 

.9456 


Table VI. 

Power q = 

a.as. 


Hollow lines. 


2. 

1 .03-2 

.4557 

.328! 

.5235 

.7111 

.86481.9652 

.5109 

.2815 

.2155 

1.226 

5328 

2.125 

.0430 

.1721 

.3557 

.5566 

.7416 

.8856 

.9731 

.5288 

.2861 

.2300 

1.248 

.5657 

2.25 

.0481 

.1887 

.3829 

.5880 

.7693 

.9243 

.9742 

.5150 

2428 

.2425 

1.272 

.5983 

2.375 

.0533 

.2055 

.4095 

.6176 

.7942 

.9L3 

.9810 

.5690 

.21-98 

.2545 

1.298 

.6164 

2.5 

.0588 

.2226 

.4356 

.6455 

.8187 

.9314 

.9876 

.5742 

.80! 6 

.2657 

1.325 

.6363 

2.625 

.0644 

.2398 

.4610 

.6716 

.8367 

.9 113 

.9904 

.58-78 

.3115 

.2762 

1.354 

.6547 

2.75 

.0703 

.2570 

.4857 

.6962 

.8547 

.9510 

.9926 

.6005 

.3158 

.2863 

1.385 

.6722 

2.875 

.0762 

.2742 

.5096 

.7191 

.8708 

.95^7 

.9943 

.6124 

.3207 

.2460 

1.413 

.6885 

’ 3. 

.0826 

.2914 

.5327 

.7405 

.8937 

.9652 

.9456 

.6235 

.3250 

.3053 

1.453 

.7085 

8.25 

.0955 

.3257 

.5766 

.7789 

.9095 

.9752 

.997 4 

■ .6 443 

.3313 

.3228 

1.524 

.7294 

3.5 

.1090 

.8595 

.6172 

.8120 

.9288 

.9825 

.9985 

.6631 

.3415 

.3388 

1.596 

.7490 

3.7 5 

.1229 

.3926 

.6547 

.8405 

.9 140 

.9876 

.9991 

.6799 

.3 480 

.3535 

1.663 

.7658 

4. 

.1373 

.4249 

.6890 

.8648 

.9560 

.9912 

.9994 

.6954 

.3543 

.3665 

1.725 

.7800 

4.5 

.1673 

.4865 

.7488 

.9035 

.973' l 

.9056 

.9998 

.7213 

.3659 

.3920 

1.883 

.8048 

5. 

.1982 

.5436 

.7981 

.9311 

.9834 

.9478 

.9999 

.7435 

.3753 

.4105 

2.038 

.8258 

6. 

.2618 

.6434 

>624 

.9652 

.9938 

.9995 

.9999 

.7794 

.3907 

.4437 

2.852 

.8622 

8. 

.3878 

.7887 

.9484 

.9412 

.99 H 

.0999 

1.000 

.8271 

.4128 

.4895 

2.988 

.8950 

10. 

.5032 

.8777 

.9797 

.9978 

.999 > 

LOCO 

1.000 

.8565 

.4210 

.5192 

3.623 

.9144 

12. 

.6029 

.9304 

.9920 

.9995 

.9999 

1.000 

1.009 

.879 4 

.4373 

.5413 

4.270 

.9315 

16. 

.7538 

.9776 

.9988 

9999 

1.000 

1.000 

1 000 

.9072 

.4524 

.5774 

5.500 

.9485 





















































































452 Parabolic Construction of Ships. 



Table VII. 

Power q = 

2.5. 


Hollow lanes 

• 

Exp. 



Ordinates. 



Prod. 

C. gr. 

Meta 

Res. 

Inflec- 

??. 

1 

2 

3 

4r 

5 

I 6 

7 


e 

m 

t 

tiou. 

2 

.0266 

.1266 

.28 19 

.4871 

.6846 

.8510 

.9611 ! 

.4)13 

.2912 

.2053 

1.248 

.5000 

2.125 

.0303 

.1415 

.3171 

.5215 

.7174 

! .8737 

.9702 

.5092 

.2954 

.2182 

1.266 

.5300 

2.25 

.0343 

.1568 

.3142 

.5543 

.7472 

.8932 

.9769 i 

.5260 

.3003 

.2310 

1.287 

.5568 

2.875 

.0384 

.1724 

.3709 

.5854 

.7741 

.9097 

.8922 

.5414 

.3084 

.2480 

1.309 

.5807 

2.5 

.0429 

.1884 

.3972 

.6149 

.8007 

.9237 

.9862 i 

.5558 

.3134 

.2545 

1.333 

.6008 

2.625 

.0474 

.2046 

.4230 

.6426 

.8203 

.9356 

.9894 j 

.5695 

.3198 

1 .2655 

1.360 

.6227 

2.75 

.0521 

.2210 

.4482 

.6688 

.8400 

.9497 

.9918 J 

.5825 

.3235 

.2762 

1.389 

.6424 

2.875 

.0572 

.2375 

.4728 

.6933 

.8o76 

.9542 

.9937 ! 

.5949 

.3286 

.2861 

1.420 

.6605 

8. 

.0626 

.2541 

.4967 

.7162 

.8742 

.9614 

.9051 jj 

.6068 

.3325 

.2954 

1.452 

.6767 

.5.25 

.0735 

2875 

.5424 

.7576 

.9000 

.9725 

.9974 | 

.6282 

.3405 

.3138 

1.520 

.7044 

3.5 

.0852 

.3209 

.5850 

.7935 

.9212 

.9806 

.9983 

.6476 

.3480 

.3296 

1.510 

.7254 

.5.75 

.0974 

.3539 

.6246 

.8244 

.9380 

.9863 

.9990 

.6650 

.3540 

.3411 

1.653 

.7432 

4. 

.1102 

.3864 

.6610 

.8510 

.9513 

.9903 

.9994 1 

.6806 

.3605 

.3577 

1.710 

.7582 

4.5 

.1371 

.4501 

.7252 

8913 

.97db 

.9951 

.9998 

.7077 

.3715 

3820 

1.861 

.7846 

.5. 

.1656 

.5080 

.7784 

.9237 

.9816 

.9976 

.9999 3 

.7311 

.3806 

.4027 

2.014 

.8070 

5. 

.2255 

6126 

.8576 

.9614 

.9931 

.9994 

1.000 | 

.7692 

.3955 

.43.65 

2.312 

.8472 

8. 

.3491 

.7682 

.9428 

.9903 

.9990 

.9999 

1.000 

.8188 

.4166 

.4835 

2.935 

.8827 

10. 

.4662 

.8651 

.9774 

.99; 6 

.9998 

1.000 

1.000 

.8496 

.4303 

.5145 

3.554 

.9029 

12. 

.5699 

.9229 

.9911 

.9995 

.9999 

1.000 

1.000 

.9732 

.4402 

.5368 

4.180 

.9223 

16. 

.7305 

.9751 

.9987 

.9999 

1.000 

1.000 

1.000 « 

.9026 

.4558 

.5738 

5.365 

.9419 


Table VIII. 


Power q 

= 3. 

Hollow Lines. 


2. 

.0129 

.0837 

.2263 

.4219 

.6347 

.8240 

.9539 

.4571 

.2739 

.1894 

1.330 

.4472 

2.125 

.0151 

.9567 

.2520 

.4579 

.6713 

.8505 

.96 13 

.4758 

.2787 

.2930 

1 341 

.4771 

2.25 

.0175 

.1082 

.2781 

.4927 

.7049 

.8732 

.9724 

.4933 

.2862 

.2151 

1.357 

.5037 

2.375 

.0200 

.1213 

.3042 

.5260 

.7355 

.8926 

.9787 

.5097 

.2927 

.2274 

1.379 

.5286 

2.5 

0229 

.1349 

.3302 

.5579 

.7651 

.9092 

.9835 

.5252 

.2970 

.2387 

1.400 

.5508 

2.625 

.0258 

.1490 

.3561 

.5882 

.7885 

.9232 

.9873 

.5397 

.3012 

.2498 

1.424 

.5723 

2.75 

.0290 

.1634 

.3817 

.6171 

.8112 

.9352 

.9902 

.5534 

.3090 

.2600 

1.450 

.5926 

2.875 

.0323 

.1782 

.1070 

.6443 

.8316 

.9453 

.9924 

.5663 

.3135 

.2702 

1.476 

.6117 

3. 

.0359 

.1932 

.4318 

.6603 

.8510 

.9538 

.9941 

.5785 

.3180 

.2798 

1.504 

.6290 

3.25 

.0136 

2241 

.4799 

.7166 

.8812 

.9675 

.9965 

.6011 

.3267 

.2973 

1.583 

.6591 

3.5 

.0520 

.2556 

.5255 

.7576 

.9062 

9767 

.9979 

.6213 

.3350 

.3136 

1.625 

.6832 

3.75 

.0611 

.2875 

.5684 

.7932 

.9231 

.9835 

.9988 

.6397 

.3420 

.3290 

1.705 

.7039 

4. 

.0709 

.3194 

.6085 

.8240 

.9419 

.9883 

.9993 

.6564 

.3482 

.3427 

1.755 

.7227 

4.5 

.0921 

.3826 

.6800 

.8734 

.9641 

.9941 

.9937 

.6355 

.3603 

.3682 

1.890 

.7514 

5. 

.1156 

.4437 

.7403 

.9092 

.9779 

.9971 

.9999 

.7102 

.3700 

.3893 

2.025 

.7750 

6. 

.1674 

.5555 

.8316 

9539 

.9917 

.9993 

1.000 

.7514 

.3860 

.4244 

2.312 

.8155 

8. 

.2828 

.7287 

.9318 

9883 

.9988 

.8999 

1.0 10 

.8042 

.4090 

.4734 

2.888 

.8635 

10. 

.4002 

.8404 

.9730 

9971 

.9099 

1.00) 

1.000 

.8378 

.4238 

.5060 

3.476 

.8845 

12. 

.5093 

.908! 

.9894 

.9993 

.9999 

1.000 

1.0)0 

.8622 

.4345 

.5293 

4.065 

.9082 

16. 

.6S68 

.9702 

.9984 

.9999 

1.000 

1.009 

1.000 

.8940 

.4490 

.5062 

5.260 

.9310 


Table IX. 

Power q = 

= 4. 

Hollow lanes. 


2. 

.0302 

.0366 

.1379 

.3164 

.5454 

.7725 

.9390 

.4063 

.2461 

.1685 

1.455 

.3778 

2.125 

.0872 

.0438 

.1592 

.3529 

.5878 

.8058 

.9527 

.4257 

.2584 

.1810 

1.461 

.4107 

2.25 

.0453 

.0516 

.1815 

.3891 

.627:1 

.8346 

.9634 

.4440 

.2607 

.1930 

1.469 

.4400 

2.375 

.0544 1 .0600 

.2045 

.4246 

.6639 

.8594 

.9717 

.4612 

.2672 

.2047 

1.478 

.4618 

2.5 

.0649 

.0692 

.2282 

.4593 

.70 >7 

.8807 

.9781 

.4774 

.2735 

.2160 

1.190 

.4865 

2.625 

.0762 

.0789 

.2524 

.4599 

.7284 

.8990 

.9831 

.4928 

.2796 

.2270 

1.505 

.5078 

2.75 

.0892 

.0893 

.2769 

.5253 

.7565 

.9145 

.9869 

.5073 

.2852 

.2374 

1.522 

.5282 

2.875 

.0130 

1024 

.3016 

.5564 

.7820 

.9275 

.9899 

.5210 

.2906 

.2475 

1.556 

.5477 

3. 

.0119 

.1117 

.3264 

.5862 

.8065 

.9389 

.9922 

.5340 

.2955 

.2570 

1.569 

.5665 

3.25 

.0154 

.1361 

.3757 

.6413 

.8449 

.9564 

.9354 

.5581 

.3046 

2747 

1.618 

.5985 

3.5 

0194 

1622 

.4211 

.6906 

.8770 

.9691 

.9972 

.5799 

.3131 ! 

.2908 

1.670 

.6261 

3.75 

.0241 

.1897 

.4709 

.7312 

.9027 

.9781 

.9 !S3 1 

.5997 

.3211 

.3056 

1.725 

.6488 

4. 

.0293 

.2184 

.5157 

.7725 

.9232 

.9845 

9990 

.6178 

.3288 

.3198 

1.787 

.6687 

4.5 

.' 416 

.2778 

.5981 

.8349 

.9524 

.9922 

.9996 

.6494 

.3415 

.3458 

1.906 

.7000 

5. 

.0563 

.3384 

.6697 

.8807 

.9707 

.9961 

.9999 

.6764 

.3529 

.3607 

2.036 

.7275 

6.5 

.0923 

.4566 

.7821 

.9390 

.9889 

.9991) 

1 .000 

.7196 

.3711 

.4957 

2.309 

.7754 

8. 

.1856 

.6558 

.9101 

.9845 

.9984 

.9999 

1.000 

.7788 

.3966 

.4575 

2 845 

.8304 

10. 

.2949 

.7931 

.9641 

.9961 

.9395 

.9999 

1.000 

.8174 

.4138 

.4924 

3.403 

.8637 

12. 

.4067 

.8796 

.9859 

.9990 

.9999 

1.000 

1.030 

.8445 

.4260 

.5174 

3.987 

.8859 

16. 

.6050 

.9605 

.9978 

.9999 

1.000 1.000 

1.000 

.8803 

.4424 

.5507 

5.087 

.9142 






















































































Parabolic Construction op Ships. 433 



Table X. 

Power q - 

0.35 


Pull Lines. 

Exp. 



Ordinates. 



I Prod. 

C. gr. 




n. 

1 

2 

3 

4: 

5 


7 

A rea. 

e 




0.25 

.4257 

.5053 

.5770 

.6316 

.6829 

.7357 

.7979 

.6126 

.4164 




0.375 

.4697 

.5655 

.6196 

.6916 

.7448 

.7979 

.‘578 

.6524 

.4210 




0.5 

.5041 

.6046 

.6765 

.7357 

.7S90 

.8408 

.8967 

.7011 

.4249 




0.625 

.5315 

.6224 

.7103 

.7700 

.8228 

.8725 

.9235 

.7317 

.4277 




0.75 

.5556 

.6637 

.7383 

.7979 

.8495 

.8967 

.9422 

.7540 

.4304 




0.875 

.5757 

.6868 

.7620 

.8202 

.8712 

.9155 

.9567 

.7721 

.4328 




1. 

.5946 

.7071 

.7825 

.8409 

.8891 

.9306 

.9672 

.7885 

.4351 




1.125 

.6111 

.7251 

.8005 

.8557 

.8039 

.9420 

.9750 

.8019 

.4323 




1.25 

.6262 

.7413 

.8164 

.8725 

.9172 

.9525 

.9809 

.8128 

.4393 




1.375 

.6398 

.7560 

.8302 

.88-34 

.9276 

.9606 

.9853 

.8224 

.4412 




1.5 

.6527 

.7694 

.8429 

.8967 

.9368 

.9672 

.9887 

.8313 

.4428 




1.75 

.6756 

.7930 

.8653 

.9156 

.9517 

.9771 

.9933 

.8463 

.4455 




2. 

.6955 

.8133 

.8835 

.9306 

.9628 

.9840 

.9961 

.8586 

.4478 




2.25 

.7137 

.8308 

.8988 

.9427 

.9413 

.9887 

.9977 

.8678 

.4499 




2.5 

.7i.99 

.8462 

.9118 

.9525 

.9780 

.9921 

.9986 

.8760 

.4519 




2.75 

.7446 

.8599 

.9229 

.9606 

.9827 

.9344 

.9992 

.8836 

.4537 




3. 

.7580 

.8720 

.9292 

.9672 

.9866 

.9361 

.9995 

.8902 

.4554 




3.5 

.7817 

.8925 

.947S 

.9771 

.9918 

.9980 

.9998 

.9023 

.4583 




4. 

.8020 

.9093 

.9594 

.9840 

.9950 

.9930 

.9999 

.9121 

.4606 




5. 

.8354 

.9345 

.9752 

.9921 

.9981 

.9997 

.9999 

.9203 

.4634 





Table XI. 

Power q = 

0.375. 

Full Lines. 

0.5 

.3579 

.4702 

.5564 

.6310 

.6693 

.7709 

.8491 

.6031 

.4003 




0.625 

.3874 

.4478 

.5986 

.6757 

.7463 

.8150 

.8875 

.6410 

.4090 




0.75 

.4141 

.5407 

.6343 

.7128 

.7830 

.8491 

.8840 

.6722 

.4150 




0.875 

.4368 

.5692 

.6652 

.7441 

.8132 

.8761 

.9358 

.6993 

.4182 




1. 

.4585 

.5940 

.6922 

.7711 

.8384 

.8977 

.9511 

.7203 

.4215 




1.125 

.6014 

.6147 

.7162 

.7945 

.8593 

.9112 

.9626 

.7388 

.4240 




1.25 

.4955 

.6383 

.7377 

.8150 

.8781 

.9296 

.9714 

.7537 

.4262 




1.375 

.5118 

.6573 

.7564 

.8331 

.8934 

.9414 

.9781 

.7673 

.4281 




1.5 

.5273 

.6749 

.7738 

.8491 

.9068 

.9511 

.9832 

.7797 

.4298 




1.75 

.5541 

.7062 

.9049 

.9761 

.9281 

.9659 

.9900 

.8000 

.4330 




2. 

.5802 

.7334 

.8305 

.8977 

.9448 

.9761 

.9941 

.8172 

.4355 




2.25 

.6030 

.7573 

.8521 

.9153 

.9572 

.9832 

.9965 

.8312 

.4381 




2.5 

.6236 

.7785 

.8707 

.9297 

.9672 

.9882 

.9379 

.8434 

.4405 




2.75 

.6425 

.7973 

.8‘66 

.9414 

.9742 

.9916 

.9992 

.8534 

.4427 




3. 

.6599 

.8142 

.8957 

.9511 

.9800 

.9941 

.9995 

.8620 

.4448 




3.25 

.5934 

.7794 

.8848 

.9460 

.9791 

.9944 

.9994 

.8693 

.4458 




3.5 

.6611 

.8432 

.9227 

.9659 

.9878 

.9971 

.9997 

.8759 

.4486 




4. 

.7183 

.8671 

.9398 

.9761 

.9925 

.9985 

.9999 

.8869 

.4523 




4.5 

.7423 

.8868 

.9523 

.9832 

.9954 

.9993 

1.000 

.8960 

.4553 




5. 

.7636 

.9034 

.9633 

.9881 

.9972 

.9396 

1.000 

.9033 

.4582 





Table XII. 


Power q 

-= 0.5. 

Full Lines. 

i. 

.3535 

.5000 

.6124 

.7071 

.7906 

.8660 

.9354 

.6666 

.3966 




1.127 

.3734 

.5258 

.64118 

.7358 

.8170 

.8873 

.9506 

.6860 

.4023 




1.25 

.3921 

.5496 

.6665 

.7613 

.8406 

.9073 

.9621 

.7048 

.4050 




1.375 

.4093 

.5716 

.6892 

.7839 

.8605 

.9227 

.9709 

.7218 

.4076 




1.5 

.4260 

.5920 

.7104 

.8040 

.8777 

.9354 

.9776 

.7382 

.4002 




1.625 

.4414 

.6111 

.7308 

.8221 

.8927 

.9496 

.9827 

.7518 

.4125 




1.75 

.4565 

.6289 

.7488 

.8383 

.9057 

.9548 

.9868 

.7646 

.4148 




1.875 

.4703 

.6457 

.7653 

.8528 

.9171 

.9621 

.9898 

.7755 

.4170 




2. 

.4841 

.6614 

.7806 

.8660 

.9270 

.9682 

.9921 

.7854 

.4192 




2.25 

.5094 

.6903 

.8079 

.8887 

.9434 

.9776 

.9953 

.8013 

.4234 




2.5 

.5327 

.7161 

.8314 

.9073 

.9565 

9842 

.9972 

.8147 

.4270 




2.75 

.5544 

.7394 

.8517 

.9227 

.9657 

.9889 

.9984 

.8262 

.4302 




3. 

.5745 

.7603 

.8634 

.9354 

9735 

.9921 

.9990 

.8376 

.4331 




3.25 

.5934 

.7794 

.8848 

.9160 

.9791 

.9944 

.9994 

.8165 

.4357 



\ 

3 5 

.6110 

.7966 

.8983 

.9526 

.9837 

.9961 

.9996 

.8554 

.4383 




3.75 

.6276 

.8124 

.9102 

9621 

.9373 

.9972 

.9998 

.8630 

.4407 




4. 

.6433 

.82(8 

.9205 

.9682 

.9900 

.9980 

.9999 

.8704 

.4430 




4.5 

.6721 

.8520 

.9377 

.9777 

.9939 

.9993 

.9999 

.8819 

.4474 




5. 

.6979 

.8733 

.9511 

.9842 

.9963 

.9995 

1.000 

.8910 

.4512 




6. 

1 _ 

.7424 

.9067 

.9697 

.9921 

.9986 

.9999 

1.000 

.9054 

.4562 







































































454 Parabolic Construction of Ships. 


Table XIII. 

Power q - 

= 0.75. 

Full Lines. 

Exp. 




Ordinates. 




C.gr. 




n 

1 

3 

3 

4 

5 

6 

7 

Area. 

e! 

Infl. 



1.5 

.2781 

.4555 

.5988 

.7209 

.8223 

.9047 

.9666 

.6594 

.3819 




1.625 

.2932 

.4777 

.6247 

.7284 

.8434 

.9201 

.9742 

.6788 

.3853 




1.75 

.3070 

.4988 

.6479 

.767-5 

.8619 

.9329 

.9802 

.6884 

.3893 




1.S75 

.3225 

.5188 

6695 

.7876 

.8782 

.9437 

.9848 

.7028 

.3929 

m 



2. 

.3367 

.5379 

.6897 

.8059 

.8925 

.9527 

.9883 

.7175 

.3963 




2.125 

.3504 

.5562 

.7085 

.8226 

.9051 

.9603 

.9909 

.7282 

.3993 

.2 m 



2.25 

.3636 

.5735 

.7261 

.8378 

.9163 

.9667 

.9930 

.7400 

.4020 

np a 

O ° 



2.375 

.3762 

.5902 

.7426 

.8516 

.9261 

.9720 

.9946 

.7504 

.4050 

o « 



2.5 

.3888 

.6060 

.7580 

.8446 

.9355 

.9765 

.9958 

.7604 

.4071 

.S'S 



2.75 

.4128 

.6358 

.7860 

.8863 

.9490 

.9834 

.0975 

.7770 

.4128 




3 

.4355 

.6630 

.8023 

.9047 

.9605 

.9S82 

.9985 

.7910 

.4168 

o 2 



3.25 

.4570 

.6880 

.8323 

.9201 

.9689 

.9917 

.9991 

.8032 

.4207 

CO <D 



•15 

.4776 

.7110 

.8514 

.9329 

.9757 

.9941 

.9995 

.8142 

.4238 

fi 33 

xi 



3.75 

.4972 

.7322 

.8683 

.9437 

.9675 

.9958 

.9997 

.8244 

.4268 

<X> 



4 

.5159 

.7518 

.8832 

.9527 

.9851 

.9971 

.9998 

.8341 

.4292 

H 



4.5 

.5510 

.7865 

.9081 

.9667 

.9909 

.9985 

.9999 

.8525 

.4333 




5 

.5830 

.8161 

.9276 

.9765 

.9944 

.9993 

.9999 

.8652 

.4372 




6 

.7178 

.8633 

.9120 

.9882 

.9979 

.9998 

1.000 

.8835 

.4451 




8 

.7292 

.9239 

.9825 

.9971 

.9997 

.9999 

1.000 

.9036 

.4562 




10 

.7954 

.9575 

.9932 

.9993 

.9998 

.9999 

1.000 

.9194 

.4623 




Table XIV. 


Power q 

=1. 



Frames. 

.125 ! 

.0165 

.0353 

.0517 

.0830 

.1154 

.1591 

.2289 

.1111 

.2647 

CO 



.2.) i 

.0320 

.0691 

.1109 

.1591 

.2171 

.2989 

.4054 

.2000 

.2777 

k 



.375 i 

.0187 

.1023 

.1616 

.2289 

.3077 

.40-74 

.5415 

.2727 

.2849 

3 

o 



.5 | 

.0646 

.1340 

.2094 

.2929 

.3876 

.5000 

.6464 

.3333 

.3000 

<p 



.625 j 

.0798 

.1646 

.2545 

.3516 

.4583 

.5795 

.7274 

.3816 

.3095 

c3 

O 



.75 i 

.0953 

.1941 

.2971 

.4054 

.5208 

.6464 

.7898 

.4286 

.3182 

3 

o 



.875 j 

.1099 

.2225 

.3372 

. 1547 

5761 

.7027 

.8379 

.4666 

.3261 

O 



1. 

.1250 

.2500 

.3750 

.5000 

■6250 

7500 

.8750 

.5000 

.3333 

strai’t 



1.125 

.1394 

.2764 

.4106 

.5415 

•6683 

.7873 

.9036 

.5294 

.3400 




1.25 

.1537 

.3020 

.444 5 

.5795 

•7065 

.8232 

.9257 

.5555 

.3461 

CO 



1.375 

.1676 

.3267 

.4 50 

.6144 

•7401 

.8513 

.9427 

.5789 

.3518 

> 



1.5 

.1815 

.3505 

.5059 

.6461 

•7704 

.8750 

.9558 

.6000 

.3571 

3 

O 



1.625 

.1948 

.3734 

.5 -41 

.6758 

•7968 

8949 

.9657 

.6190 

.3621 

X 



1.75 

.2084 

.3955 

.5607 

.7027 

•8203 

.9116 

.9737 

.6363 

.3666 

k 

a 



1.875 

.2211 

.4169 

.5857 

.7274 

•8410 

.9257 

.9797 

.6522 

.3710 

© 

o 



Table XV 

• 


Power q 

= 1.25. 


Frames. 

.125 

.( 059 

.0137 

.0279 

.0446 

.0671 

.1005 

.1583 

.0765 

.2120 

S3 



.255 

.0140 

.0329 

.0640 

.1005 

.1485 

.2155 

.3235 

.1500 

.2421 

t a - 



.375 

.0229 

.0578 

.1024 

.1582 

.2292 

1.3242 

.4645 

.2222 

.2555 

w 

<8.2 



.5 

.0326 

.0808 

.1417 

.2155 

.3059 

.4201 

.5796 

.2755 

.2687 

• S g 



.615 

.0424 

.1048 

.1808 

.2707 

.3771 

.5054 

.6717 

.3264 

.2800 

V « 



7 - 

.0530 

.1288 

.2193 

.3235 

.4424 

.5797 

.7445 

.3691 

.2895 

c3 



.875 

.0633 

.1528 

.2569 

.3734 

.5019 

.6434 

.8095 

.4080 

.2991 

g § 



.1 

.0743 

.1768 

.2934 

.4204 

.5557 

.6980 

.8463 

.4432 

.3073 

o 



1.125 

.0852 

.2004 

.3287 

.4645 

.6033 

.7417 

.8810 

.4755 

.3145 

.3332 



1.125 

.0963 

.2239 

.3627 

.5056 

.6478 

.7841 

.9080 

.5040 

.3212 

.5135 



1.375 

.1072 

.2470 

.3943 

.5410 

.6868 

.8178 

.9289 

.5299 

.3273 

.6175 



1.5 

.1185 

.2697 

.4254 

.5796 

.7217 

.8463 

.9451 

.5523 

.3333 

.6886 



1.625 

.1294 

.2919 

.4566 

.6127 

.7529 

.8708 

.9574 

.5726 

.3386 

.7382 



1.75 

.1408 

.3137 

4'52 

.6434 

.7807 

.8908 

.9673 

.5910 

.3435 

.7710 



1.875 

.1516 

.3350 

.5124 

.6717 

.8054 

.9080 

.9747 

.6084 

.3483 

.7966 







































































Parabolic Construction of Ships. 


Table XVI. 


Power q = 1.1 

5. 


Frames. 

Exp. 



Ordinates. 




C. gr. 




n 

1 

3 

3 

4 

5 

6 

7 

Area. 

e 

Inti. 



.125 

.0021 

.0058 

.0135 

.0239 

.0391 

.0635 

.1095 

.0650 

.1915 

A 



.25 

.0059 

.0166 

.0369 

.06:55 

.1014 

.1585 

.2581 

.1252 

.2125 

£ d 



.375 

.0107 

.0327 

.0818 

.1094 

.1707 

.2581 

.3985 

1810 

.2290 

CO o 



.5 

.0164 

.0489 

.0958 

.1585 

.2413 

.3532 

.5197 

.2313 

.2423 

© O 
© 



.625 

.0225 

.0668 

.1284 

.2085 

.3102 

.4412 

.62 33 

.2792 

.2552 

%% 



.75 

.0294 

.0^55 

.1619 

.2581 

.3758 

.5197 

.7019 

.3238 

.2658 

g g 



.875 

.0364 

.1050 

.1058 

.3067 

.4372 

.5890 

.7670 

.3638 

.2763 

a a 
© 



1. 

.0442 

.1250 

.2296 

.3535 

.4940 

.6489 

.8185 

.4000 

.2850 

o 



1.125 

.0521 

.1453 

.2632 

.3985 

.5454 

.6986 

.8590 

.4328 

.2937 

.2068 



1.25 

.0603 

.1660 

.2961 

.4412 

.5939 

.7469 

.8906 

.4626 

.3015 

.3333 



1.375 

.0686 

.1867 

.3274 

.4816 

.6371 

.7855 

.9153 

.4892 

.3091 

.4441 



1.5 

.0773 

.2075 

.3586 

.5197 

.6761 

.8185 

.9344 

.5131 

.3160 

.5429 



1.625 

.0860 

.2282 

.3903 

.5555 

.7113 

.8465 

.9491 

.5348 

.3224 

.6115 



1.75 

.0451 

.2488 

.4198 

.5890 

.7429 

.8701 

.9608 

.5545 

.32S5 

.6534 



1.875 

1 .1040 

.2692 

.44S3 

.6203 

.7713 

89)6 

.9698 

.5727 

.3342 

.6838 



Table XVII. 


Power q 

= 1.75. 


Frames. 

.125 

.0008 

.0025 

.0067 

.0128 

.0228 

.0401 

.0758 

.0500 

1.1623 

A 



.25 

.0025 

.00S4 

.0213 

.0401 

.0693 

.1166 

.2060 | 

.1000 

.1876 

* a 



.375 

.0050 

.0185 

.0412 

.0767 

.1272 

.2060 

.44 >3 

.1482 

.2058 

” 2 



.5 

.00 S3 

.0296 

.0648 

.1166 

1904 

.3030 

.4660 

.1955 

.2200 

a © 

■A V 



.625 

.0120 

.0425 

.0912 

.1605 

.2553 

.3850 

.5729 

.2418 

.2337 

si 



.75 

.0164 

.0767 

.1195 

.2060 

.3193 

.4660 

.6617 

.2860 

.2156 

1 g 



.875 

.0210 

.0721 

.1492 

.2518 

.3809 

.5393 

.7338 

.3272 

.2563 

a g 

o 



1. 

.0263 

.08-'4 

.1797 

.2973 

.4393 

.6044 

.7916 

.3636 

.2663 

O 



1.125 

.0318 

.1054 

.2107 

.3418 

.4929 

.6581 

.8375 

.3973 

.2757 

.1619 



1.25 

.03:7 

.1231 

.2118 

.3850 

.5445 

.7115 

.8736 

.4276 

.2848 

.2888 



1.375 

.0439 

.1412 

.2718 

.4264 

.5910 

.7545 

.9019 

.4556 

.2937 

.3845 



1.5 

.0506 

.1596 

.3022 

.4660 

.6334 

.7916 

.9240 

.4796 

.3008 

.4575 



1.625 

.0571 

.1784 

.3337 

.5037 

.0721 

.8234 

.9409 

.5015 

.3072 

.5193 



1.75 

.0643 

.1973 

.3633 

5393 

.7070 

.8505 

.9545 

.5222 

.3137 

.5676 



1.875 

.0713 

.1718 

.3922 

.5729 

.7386 

.8736 

.9648 

.5414 

.3200 

.6052 



Table XVIII. 


Power q 

L = 3. 



Frames. 

.125 

.0003 

.0011 

.0033 

.0069 

.0133 

.0253 

.0524 

.0222 

.1470 

A 

s-> 



.25 

.0011 

.0043 

.0113 

.0253 

.0473 

.0858 

.1643 

.0666 

.1666 

£ d 



.375 

.0024 

.0105 

.0261 

.0523 

.9471 

.1643 

.2932 

.1168 

.1842 

w .2 
© 



.5 

.0042 

.0179 

.0430 

.0858 

.1503 

.2497 

.4179 

.1666 

.2000 

.2 8 



.625 

.0064 

.0271 

.0648 

.1236 

.2100 

.3359 

.5323 

.2136 

.2145 

© a 



.75 

.0091 

.0377 

.0652 

.1643 

.2712 

.4179 

.6237 

.2571 

.2273 

1 a 



.875 

.0121 

.0495 

.1137 

.2068 

.3319 

.4938 

.7021 

.2969 

.2395 

© 



1. 

.0156 

.0625 

.1496 

.2500 

.3906 

.5625 

.7656 

.3333 

.2500 




1.125 

.0194 

.0764 

.1686 

.2932 

.4456 

.6199 

.8165 

.3665 

.2600 

.1291 



1.25 

.0236 

.0912 

.1974 

.3359 

.4992 

.6777 

.8569 

.3968 

.2691 

.2350 



1.375 

.0281 

.1067 

.2256 

.3775 

.5482 

.7248 

.SS87 

.4245 

.2780 

.3244 



1.5 

.0330 

.1128 

.2547 

.4179 

.5934 

.7656 

.9136 

.4500 

.2860 

.3968 



1.625 

.<'380 

.1394 

.2853 

.4567 

.6350 

.8008 

.9317 

.4734 

.2930 

.4587 



1.75 

.0434 

.1565 

.3144 

.4938 

.6729 

.8310 

.9481 

.4949 

.3000 

.5092 



1.875 

.0189 

.1738. 

•3431 

.5291 

.7073 

.8569 

.9593 

.5149 

.3062 

.5473 



Table XIX. 


Power q 

= 3.25. 


Frames. 

.375 

.0011 

.0059 

0166 

.0362 

.0705 

.1311 

.2515 

.1022 

.1237 

<5 d 




.0021 

.0108 

.0297 

.0631 

.1213 

.2099 

.3747 

.1441 

.1546 

a o 



.625 

.0060 

.0172 

.0460 

.0952 

.1728 

.2931 

.4886 

.1863 

.1629 

<U 8 



.75 

.0051 

.0251 

.0481 

.1342 

.2304 

.3747 

.5881 

.2270 

.1855 

g 2 



.87-5 

.0069 

.0340 

.0866 

.1698 

.2891 

.4521 

.6717 

.2675 

.1990 

o g 



1. 

.0045 

.0442 

.1100 

.2102 

.3473 

.5235 

.7405 

v 1 8 

.2110 




1.125 

0119 

.9554 

.1350 

.2515 

.4027 

.5839 

.7961 

.3412 

.2216 

.1078 



1.25 

.0148 

.0676 

.1612 

.2931 

.4577 

.6455 

.8405 

.3717 

.2331 

.2015 



1.375 

.0180 

.0807 

.1839 

.3343 

.5085 

.6962 

.8757 

.4000 

.2422 

.2833 



1.5 

.0215 

.0945 

.2147 

.3747 

.5560 

.7405 

.9033 

.4251 

.2512 

.3515 



1.625 

.0252 

.1090 

:2439 

.4141 

.6000 

.7789 | 

.9246 

.4:89 

.2590 

.4032 






































































































456 


Parabolic Construction or Siik. 


---——j 

Table XX.—For Elliptic Stem of Vessels. 


Ex¬ 

po- 


Ordinates of Ellipses of Different Order. 

Area 

Index 

neut 









ST —- 

i 

n. 

% 

1 

3 

3 

4 

5 

G 

7 

bl X 


2. 

.3398 

.4840 

.6616 

.7808 

.8660 

.9204 

.9682 

.9922 

.7854 

1.603 

2.25 

.4108 

.5490 

.7147 

.8274 

.9004 

.9495 

.9801 

.9958 

.8154 

1.544 

2.5 

.4670 

.6042 

.7657 

.8627 

.9252 

.9546 

.9873 

.9978 

.8382 

1.495 

2.75 

.5174 

.6514 

.8029 

.8901 

.9434 

.9749 

.9932 

.9989 

.8564 

1.453 

3. 

.5604 

.6911 

.8331 

.9019 

.9565 

.9821 

.9948 

.9994 

.8709 

1.418 

3.25 

.5991 

.7252 

.8578 

.9275 

.9664 

.9871 

.9973 

.9996 

.8833 

1.388 

3.5 

.6333 

.7548 

.8782 

.9406 

.9740 

.9907 

.9978 

.9998 

.8935 

1.362 

4. 

.6906 

.8021 

.9003 

.9595 

.9840 

.9950 

.9995 

.9999 

.9075 

1.318 

d f 

30° 

.0149 

.0582 

.1311 

.2374 

.3740 

.54491 

.7531 

) Sheer 


45° 

.0157 

.0539 

.1221 

.2152 

.3517 

.5227 

.7313 

r of 


60° 

.0160 

.0474 

.1086 

.1972 

.3190 

.4794 

.6946 

\ vessels. 


Table XXI.—To Approximate Size and Shape of Vessels. 




d 

n ; 



Exponent for displacement 

n. q 

r-2. 


3 

3.5 

3 

3.5 

4 

5 

6 

8 

10 

.356 

.397 

.429 

.453 

.474 

.500 

.528 

.558 

.577 

.381 

.425 

.459 

.486 

.508 

.541 

.566 

.597 

.620 

.400 

.447 

.482 

.510 

.533 

.563 

.594 

.627 

.650 

.414 

.462 

.500 

.528 

.552 

.589 

.616 

.650 

.673 

.427 

.476 

.514 

.544 

.569 

.606 

.635 

.668 

.693 

.444 

.496 

.535 

.567 

.592 

.631 

.660 

.696 

.722 

.458 

.509 

.550 

.583 

.610 

.649 

.679 

.717 

.750 

.474 

.529 

.571 

.605 

.632 

.673 

.704 

.743 

.770 

.490 

.547 

.590 

.625 

.654 

.696 

.720 

.759 

.797 

Speed and 

Freight and 

Freight and 

passengers. 

passengers. 

slow speed. 



Purpose. 


Table XXII.—Length of Vessels=Tab«lar Number 2'. 


Coefficient C. 


Propc 

rtion 

of draft and 

length of vessels 




8 

13 

18 

36 

36 

48 

64 

83 

103 

Il 

.356 

14.9 

19.0 

23.7 

28.9 

34.0 

38.9 

43.6 

47.2 

48.0 

•H O V 

d d ^ 

.425 

13.9 

17.7. 

22.1 

26.9 

31.7 

36.2 

40.6 

43.9 

44.5 


.482 

13.3 

17.0 

21.2 

25.8 

30.3 

34.7 

38.9 

42.1 

42.7 


f .528 

12.9 

16.5 

20.5 

25.0 

29.4 

33.7 

37.7 

40.8 

41.3 

f 1^ 

.569 

12.5 

16.0 

20.0 

24.4 

28.7 

30.8 

36.8 

39.8 

40.3 

© 

.631 

12.1 

15.5 

19.4 

23.5 

27.6 

31.6 

35.4 

38.3 

38.8 


' .679 

11.8 

15.1 

18.8 

22.9 

26.9 

30.8 

34.6 

37.4 

38.0 

SB -< 

® D 

.723 

11.6 

14.8 

18.5 

22.5 

26.5 

30.3 

34.0 

36.8 

37.2 


1.797 

11.2 

14.3 

17.9 

21.8 

25.6 

29.3 

32.9 

35.6 

36.0 

Condition. 

Vessels for 
deep watir. 

Ordinary 

navigation. 

Fiver steamers, 
light draft. 



























































































Plate VI 1. 













































































































































































































































































































































































































































J - 








' * 




































































































. 


















































































Plate VIII. 






























































































Parabolic Construction of C, 



J. W.JSfystrorn 




































































































































l 
























* 













































.« 


























1 ystnnn's Parabolic / on.si no lion of Pips. 























































































































































































































































































. 








































































































To Construct a Displacement Scale. 


157 


TO CONSTRUCT A DISPLACEMENT SCALE. 

D = displacement of the vessel in cubic feet. 

6 — displacement in cubic feet per inch of difference of draft, 
a = area of load water line in square feet, d = draft of water in feet. 
a = area of any water line at draft y and displacement x. 
n = exponent of the displacement scale. 


a d 


D 


Mil). 

d" 

II 

Dy n ~\ 


d n 

12 d n 



Example. The area of the load water line of a vessel is a = 6400 square feet 
draft of water d — 17 teet, and the load draft displacement D = 80,500 cubic feet 
Required the draft expouent n — ? and at what draft y the displacement is x 
45,000 cubic feet ? 

U = 64 «n^n 17 ' ~ 1 - 35 > V = 17 i/^5000 = n.o5 feet, the draft required. 
80500 80500 


Construct a scale as shown by the accompanying figure, and draw the ordi¬ 
nates x ; the draft d being divided into eight equal parts. 

Assuming the displacement as unit, the ordinates x are found in the following 
table, opposite the given exponent n. 

After the exponent is known, the displacement can be expressed in tons, and 
the load draft displacement multiplied by the tabular number gives the displace¬ 
ment x at the corresponding draft y. 


Rule, Multiply the load draft displacement, expressed either in tons or cubic 
feet, by the tabular number for the given exponent and water line, and the pro¬ 
duct is the corresponding displacement. 


Displacement Scale. 


_ a d 

~ D 

1 

2 

Ordim 

3 

ite Wat 

4: 

erlines. 

5 

6 

7 

Dead rise. 

1.00 

.1250 

.2500 

.3750 

.5000 

.6250 

.7500 

.8750 

Elat bottom. 

1.05 

.1127 

.2333 

.3571 

.4830 

.6105 

.7393 

.8692 


1.10 

.1015 

.2476 

.3300 

.4665 

.5963 

.7287 

.8634 


1.15 

.0915 

.2031 

.3237 

.4506 

.5824 

.7183 

.8577- 


1.20 

.0825 

.1895 

.3082 

.4353 

.5689 

.7080 

.8512 

jO 

1.25 

.0743 

.1768 

.2935 

.4205 

.5557 

.6980 

.8463 

C £ 

1.30 

.0669 

.1649 

*.2794 

.4061 

.5428 

.6880 

.8407 

£*2 

1.35 

.0604 

.1539 

.2660 

.3923 

.5302 

.6782 

.8351 

o 2 

1.40 

.0544 

.1436 

.2533 

.3789 

.5179 

.6685 

.8295 

g ® 

1.45 

.0490 

.1340 

.2303 

.3660 

.5047 

.6589 

.8240 

cd +2 

1.50 

.0447 

.1250 

.2297 

.3535 

.4941 

.6495 

.8185 

5.2 

1.55 

.0398 

.1166 

.2187 

.3415 

.4826 

.6402 

.8130 


1.60 

.0359 

.1088 

.2082 

.3299 

.4714 

.6311 

.8076 


1.65 

.0323 

.1015 

.1982 

.3186 

.4605 

.6221 

.8023 

si) 

1.70 

.0291 

.0947 

.1887 

.3078 

.4498 

.6132 

.7969 


1.75 

.0257 

.0884 

.1797 

.2985 

.4393 

.6044 • 

.7916 


1.80 

.0233 

.0824 

.1711 

.2872 

.4291 

.5958 

.7864 

-t—< 

1.85 

.0213 

.0769 

.1629 

.2774 

.4129 

.5873 

.7811 


1.90 

.0192 

.0718 

.1551 

.2680 

.4094 

.5789 

.7759 


1.95 

.0173 

.0670 

.1477 

.2588 

.4008 

.5706 

.7708 

Highest 

2.00 

.0156 

.0625 

.1406 

.2500 | 

.3906 

.5625 

.7656 

dead rise. 


























































4o8 


Approximate Lengths of Vessels. 


a 

a s 


Sharp 

Vessels, ffi = 

■- 3. 

I)nq = 

2X2. 


C/ o 

s s 


Length L, Beam B, 

and draft d 

= all in 

feet. 


p<.S 

I. = 

L = 

L = 

L = 

L = 

L = 


Displace- 

L = 

L = 

I. — 

L = 

L = 

L == 

■p 

51} 

6B 

7B 

8B 

OB 

10B 


ment. 

6B 

6B 

7B 

8B 

9B 

10 B 

T 

L, 

4, 

4 

4 

4 

4 


T 

L, 

I, 

Ij 

L. 

L, 

Jj 

1 

16-6 

18-3 

203 

22-2 

240 

25-8 


1000 

166 

183 

203 

222 

240 

2-8 


210 

22-9 

25-6 

28-0 

30-3 

32 5 


1100 

171 

189 

210 

230 

248 

267 

3 

240 

26-4 

29-3 

32-0 

346 

37 2 


1200 

177 

194 

216 

236 

255 

274 

4r 

264 

29-0 

32 3 

35-3 

38-2 

41-0 


1300 

181 

199 

222 

242 

262 

2 SI 

5 

28-4 

31-4 

34-8 

38-0 

42-0 

44-2 


1400 

186 

204 

2.6 

249 

269 

288 

6 

30-3 

33-3 

37-0 

40-4 

43-8 

47-0 


1500 

190 

210 

233 

-oh 

275 

295 

7 

31 7 

34-9 

38-8 

42-4 

46-0 

49-3 


1000 

195 

214 

238 

260 

281 

302 

8 

33-2 

36-6 

40-6 

444 

48-0 

51-4 


1700 

199 

218 

243 

265 

286 

;o^ 

9 

34-6 

38 0 

42-2 

46-2 

50-0 

53-7 


1800 

202 

223 

248 

270 

292 

314 

10 

35-7 

39-3 

43-7 

47-8 

51-7 

55-5 


1900 

206 

227 

252 

275 

298 

320 

11 

36-9 

40-6 

45-1 

49-3 

53-4 

57-3 


2000 

210 

230 

256 

280 

302 

325 

1M 

38-0 

41 9 

46-6 

50-9 

5o"0 

591 


2100 

212 

234 

260 

2 i h 

307 

330 

13 

39 0 

43-0 

47-8 

52-2 

56 5 

60-6 


2200 

216 

238 

265 

290 

312 

3->5 

1+ 

400 

44-0 

49 0 

53-6 

58-0 

62-2 


2300 

220 

242 

269 

295 

317 

341 

15 

40-9 

45-0 

50-0 

54 7 

69-0 

63-5 


2400 

223 

215 

272 

300 

322 

346 

16 

41-8 

46 0 

51-2 

56-0 

60-6 

65-0 


2500 

226 

248 

276 

304 

326 

351 

17 

42-7 

46-9 

52-2 

57 0 

61-8 

663 


2600 

229 

252 

280 

308 

330 

355 

18 

43-G 

47-8 

53-2 

58-0 

63 0 

67-7 


2700 

232 

255 

283 

311 

384 

360 

19 

44-4 

48-6 

54'2 

590 

642 

690 


2800 

235 

259 

287 

315 

338 

365 

20 

45-0 

49-5 

55-1 

60 0 

65-2 

70-0 


2900 

238 

261 

290 

318 

342 

369 

25 

48-5 

534 

59-2 

64-8 

70-2 

75-4 


3000 

240 

264 

294 

321 

347 

373 

30 

51 6 

57-0 

63-3 

69 0 

74-7 

80-2 


3100 

243 

267 

297 

324 

350 

377 

35 

543 

59-6 

66-4 

72-7 

78 6 

843 


3200 

246 

270 

300 

327 

354 

381 

40 

56-8 

62-6 

69 5 

760 

82-2 

88-3 


3300 

248 

272 

303 

330 

358 

385 

45 

59-2 

650 

72-3 

79-0 

85 - 5 

92-0 


3400 

250 

275 

306 

333 

362 

339 

50 

61-2 

67-2 

748 

81-8 

88'4 

95-0 


3500 

253 

278 

309 

337 

365 

393 

55 

63-1 

69-4 

77-3 

84-4 

914 

98-0 


3000 

255 

281 

312 

340 

369 

396 

60 

64-9 

71-4 

79-4 

868 

93-9 

101 


3700 

257 

283 

314 

344 

372 

399 

65 

GG 8 

73-5 

81-7 

89-3 

96-6 

104 


3800 

260 

285 

317 

847 

375 

403 

70 

68-4 

75-3 

83-7 

91-6 

98-0 

107 


3900 

262 

288 

320 

350 

878 

406 

75 

704 

77-1 

85-8 

94-8 

101 

109 


4000 

265 

291 

3z3 

353 

380 

409 

SO 

71-6 

78-9 

87-6 

95-7 

103 

111 


4100 

268 

293 

325 

356 

o!-3 

413 

85 

73-0 

805 

895 

97-7 

106 

114 


4200 

270 

296 

328 

359 

386 

417 

90 

745 

819 

911 

99 6 

108 

116 


4300 

272 

298 

331 

362 

389 

420 

95 

75-8 

834 

92 7 

111 

100 

118 


4400 

273 

300 

334 

365 

392 

423 

100 

77-0 

84-7 

944 

103 

112 

120 


4500 

275 

302 

337 

368 

395 

426 

110 

79-7 

87 - 6 

97-4 

lu7 

115 

124 


4G00 

277 

304 

339 

370 

398 

429 

1:45 

83 0 

915 

100 

111 

120 

129 


4700 

279 

306 

331 

372 

401 

432 

150 

88'3 

97-0 

108 

113 

128 

137 


4800 

281 

308 

343 

374 

404 

435 

175 

930 

102 

114 

124 

134 

144 


4900 

283 

310 

845 

376 

407 

438 

200 

973 

107 

119 

130 

140 

151 


5000 

286 

312 

318 

378 

411 

441 

225 

101 

111 

124 

135 

146 

157 


5250 

289* 

318 

354 

386 

418 

450 

250 

105 

115 

128 

140 

152 

163 


5500 

294 

323 

359 

392 

424 

457 

275 

108 

119 

132 

145 

156 

168 


5750 

298 

331 

364 

398 

430 

463 

300 

111 

122 

136 

149 

161 

173 


6000 

303 

336 

370 

401 

437 

469 

325 

114 

126 

140 

153 

165 

178 


6250 

307 

340 

375 

409 

443 

47 6 

350 

117 

129 

143 

157 

170 

182 


6500 

310 

345 

380 

414 

418 

482 

375 

120 

130 

147 

161 

173 

186 


6750 

315 

349 

384 

420 

454 

488 

400 

122 

135 

150 

164 

177 

191 


7000 

319 

354 

389 

425 

460 

494 

450 

128 

140 

156 

170 

184 

198 


7250 

322 

358 

394 

430 

465 

500 

500 

132 

145 

161 

176 

191 

205 


7500 

326 

361 

398 

435 

470 

506 

550 

13G 

150 

166 

182 

197 

211 


7750 

330 

365 

402 

440 

475 

512 

600 

140 

154 

171 

188 

202 

217 


8000 

333 

370 

407 

445 

480 

517 

650 

144 

158 

176 

193 

208 

223 


8250 

337 

374 

410 

449 

485 

522 

700 

148 

162 

180 

197 

213 

229 


8500 

340 

377 

415 

453 

490 

527 

750 

151 

166 

185 

202 

218 

235 


875 0 

343 

380 

419 

457 

495 

532 

800 

154 

170 

189 

206 

223 

240 


9000 

347 

384 

423 

461 

500 

537 

850 

158 

173 

193 

210 

227 

245 


925 0 

350 

388 

427 

466 

504 

542 

900 

160 

176 

196 

215 

232 

250 


95 CO 

353 

392 

430 

470 

509 

547 

950 

163 

1 0 

200 

219 

236 

254 


10000 

359 

399 

438 

478 

517 

556 
































Approximate Lengths op Vessels. 450 


Pisplacem’mt 
in tons. 

L = 
5B 

i* 

Lesig 

L = 
6B 

call 

L = 
7B . 

1 v< 

Li, E 

L = 
8B 

esst 

team 

L = 
9B 

ilS. 

R, a 

L = 

10 B 

Mn == 

nd draf 

Displace¬ 

ment. 

16. 
t d = 

h = 
5B 

Dnc 

= |B 

L = 

6B 

1 = < 
, all 

L = 
7B 

>X^ 

in I 

L = 
8B 

L. 

Feet. 

L = 
9B 

L = 
10B 

T 

L. 

t. 

L. 

L. 

L, 

Li 

T 

L 

I, 

L 

L 

Li 

L, 

1 

10-9 

146 

16-2 

17-8 

19“2 

20-6 

1000 

109 

146 

162 

178 

192 

206 


13-7 

18-3 

20-4 

22-4 

24-2 

26-0 

1100 

112 

151 

166 

184 

199 

213 

3 

15-7 

21-0 

23-4 

25-7 

27-7 

29-7 

1300 

116 

155 

173 

190 

205 

219 

4 

17-3 

23-2 

25-8 

28-3 

30-5 

32-8 

1300 

119 

160 

177 

195 

210 

225 

5 

18-7 

25-0 

27-7 

30-4 

32-8 

35-2 

1400 

122 

164 

ISO 

199 

216 

230 

6 

19 8 

26'6 

29 5 

32-4 

34-9 

37-6 

1500 

1-25 

167 

184 

204 

221 

235 

7 

20-8 

28-0 

31-0 

340 

36-7 

39-4 

1600 

127 

171 

188 

208 

226 

240 

8 

21-8 

29-2 

32 4 

35 6 

38-4 

41-2 

17 00 

130 

174 

193 

212 

230 

245 

9 

22-7 

30-4 

33-7 

37-0 

40-0 

428 

1800 

132 

177 

198 

217 

235 

250 

1(1 

23-5 

31-4 

349 

38-3 

41-3 

44-3 

1900 

135 

181 

201 

220 

239 

255 

11 

242 

324 

36-0 

39 5 

42-6 

45-7 

3000 

137 

184 

205 

224 

243 

260 

13 

25-0 

333 

37 1 

40-7 

440 

47-2 

3100 

139 

188 

208 

228 

247 

264 

13 

25’6 

34-3 

381 

41-8 

45-1 

48-4 

3300 

141 

190 

211 

231 

251 

268 

14 

26-3 

35-2 

39-0 

429 

46-2 

49-6 

3300 

144 

193 

214 

234 

254 

272 

15 

26-8 

36-0 

39 9 

43-8 

47-2 

50 7 

3400 

146 

195 

217 

238 

258 

276 

16 

27-5 

36 8 

40-9 

448 

48-4 

51-8 

3500 

148 

198 

220 

242 

261 

280 

17 

28-0 

37 5 

41-8 

45 - 7 

49-4 

529 

3600 

150 

200 

223 

245 

264 

284 

18 

28-5 

38 3 

42-6 

466 

50-3 

540 

3700 

152 

203 

226 

248 

267 

287 

19 

29-0 

39 0 

43-4 

47 4 

513 

56-0 

3800 

154 

205 

228 

251 

270 

290 

30 

29 5 

39-6 

44-2 

48-2 

521) 

58-0 

3900 

155 

208 

231 

254 

274 

294 

35 

31-8 

42-6 

45’4 

52 0 

56-1 

60-0 

3000 

157 

211 

234 

257 

277 

297 

30 

33-9 

45-5 

50-5 

55-3 

599 

64-0 

3100 

159 

213 

237 

260 

280 

300 

35 

35-6 

47-8 

53 1 

58-1 

62 9 

67-2 

3300 

160 

215 

240 

262 

284 

304 

40 

373 

50 0 

55-6 

608 

65-7 

70-4 

3300 

162 

218 

242 

264 

287 

307 

45 

38-8 

52-0 

57-8 

633 

68 4 

73-3 

3400 

164 

220 

245 

267 

290 

310 

50 

401 

53-8 

59-8 

65-5 

70-7 

75-7 

3500 

166 

222 

248 

270 

292 

313 

55 

414 

5 5 '5 

61-7 

67-7 

73-1 

78-2 

3600 

167 

225 

250 

272 

295 

316 

60 

42-6 

57-1 

635 

69-5 

75-2 

80-4 

3700 

169 

226 

452 

275 

297 

319 

65 

43-8 

58-7 

65-2 

71-5 

77-4 

82-7 

3800 

171 

228 

254 

278 

299 

322 

70 

45 0 

60-2 

66-9 

73-4 

79-2 

85 0 

3900 

172 

230 

255 

280 

302 

325 

75 

46-0 

61-6 

68-6 

75-2 

812 

87-0 

4000 

173 

232 

258 

283 

305 

327 

80 

47-0 

63-0 

70 0 

770 

82-9 

88-8 

4100 

174 

234 

261 

285 

308 

330 

85 

47-9 

64-2 

71-5 

78-2 

84-6 

905 

4300 

175 

236 

263 

287 

310 

333 

90 

48 8 

65 5 

72-8 

79-7 

86 3 

92-3 

4300 

177 

238 

265 

2S9 

312 

336 

95 

49-7 

66-6 

742 

81-3 

87-8 

94-0 

4400 

178 

240 

267 

291 

314 

338 

100 

50-5 

67 8 

75-4 

82-7 

89-2 

95-6 

4500 

179 

242 

269 

294 

317 

340 

. 110 

52-2 

70-0 

78-0 

85‘3 

92-2 

98-5 

4600 

181 

244 

271 

296 

320 

343 

135 

55-0 

73-0 

810 

89-0 

95-0 

103 

4700 

182 

246 

273 

298 

322 

345 

150 

57-8 

77 : 5 

86-4 

94-5 

102 

110 

4800 

184 

248 

275 

300 

324 

347 

175 

60-8 

816 

90-8 

99-5 

108 

115 

4900 

185 

249 

276 

302 

326 

350 

300 

63-6 

85-4 

952 

104 

113 

120 

5000 

187 

250 

277 

304 

329 

352 

335 

66 2 

88-9 

99-0 

108 

117 

125 

5350 

190 

254 

278 

309 

334 

354 

350 

68-7 

92 0 

102 

112 

121 

130 

5500 

193 

258 

287 

315 

339 

356 

375 

70-8 

95-0 

106 

116 

125 

134 

5750 

196 

262 

291 

319 

345 

369 

300 

72 8 

97-8 

109 

119 

129 

138 

6000 

198 

266 

295 

324 

350 

374 

335 

74-0 

ioo 

112 

123 

132 

142 

6350 

200 

269 

300 

328 

354 

383 

350 

76-8 

101 

115 

126 

136 

146 

6500 

203 

273 

303 

332 

359 

384 

375 

78-7 

106 

118 

129 

139 

149 

6750 

206 

276 

307 

336 

364 

390 

400 

80-3 

108 

120 

132 

142 

152 

7000 

209 

280 

311 

340 

368 

394 

450 

85-5 

112 

125 

137 

148 

158 

7350 

211 

283 

314 

344 

372 

399 

500 

86-4 

116 

130 

142 

153 

164 

7500 

213 

286 

318 

348 

376 

404 

550 

89-2 

120 

133 

146 

158 

169 

7750 

216 

289 

312 

352 

380 

407 

600 

91 9 

123 

136 

150 

162 

174 

8000 

218 

292 

325 

356 

384 

412 

650 

94-4 

127 

140 

154 

167 

178 

8350 

220 

2D6 

328 

360 

388 

417 

700 

96-6 

130 

144 

158 

170 

183 

8500 

222 

299 

331 

364 

392 

421 

750 

98-9 

133 

148 

162 

173 

187 

8750 

224 

301 

334 

367 

396 

425 

800 

101 

136 

151 

165 

177 

191 

9000 

226 

304 

337 

370 

400 

429 

850 

103 

139 

154 

169 

182 

195 

9350 

229 

307 

310 

374 

404 

432 

900 

105 

142 

157 

172 

186 

199 

9500 

241 

310 

314 

377 

408 

437 

950 

107 

144 

159 

175 

1 189 

203 

10000 

1 235 

315 

350 

381 

414 

446 


























































460 


Steamship Performance, 


! 

Horsepower in Steamship Performance. 


Displace¬ 
ment in 



Nautical miles or knots per hour. 



tons. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

T 

11 

ii 

ii 

H 

H 

II 

H 

II 

ii 

II 

1 

0004 

0-035 

0-118 

0-280 

0-550 

0-949 

1-50 

2-24 

3-20 

4-38 

2 

0-007 

0-055 

0-190 

0-444 

0-870 

1-51 

2-40 

3-55 

5-08 

6 96 

3 

0‘009 

0-075 

0-248 

0-59S 

1 14 

1-98 

3-12 

4-79 

6-91 

9-12 

4: 

o-oio 

0-084 

0 300 

0-673 

1-40 

2-40 

3-80 

5-39 

8 06 

11-10 

5 

0-012 

0-102 

0-348 

0-818 

1-52 

2-78 

4 40 

6-55 

9-36 

122 

6 

0‘014 

0115 

0390 

0-924 

1-81 

3-12 

4-96 

7-39 

10 6 

14-5 

7 

0-016 

0-128 

0-435 

1-025 

2-01 

3 48 

550 

8-20 

11-7 

161 

8 

0-017 

0138 

0-479 

1125 

2 20 

3-80 

6-01 

8-96 

12-8 

175 

9 

0-019 

0-151 

0-501 

1-211 

2-38 

4-12 

6-51 

9-69 

13-8 

190 

10 

0020 

0-161 

0-552 

1-30 

2-54 

4-42 

698 

10-4 

149 

20-3 

11 

0-022 

0175 

0590 

1-40 

2-72 

4-70 

7-46 

111 

15-9 

21-8 

12 

0-023 

0185 

0 624 

1-48 

2-88 

4-99 

7-90 

11-8 

16-8 

23-0 

13 

0-024 

0-195 

0-654 

1-56 

3-04 

5-25 

8-33 

12-5 

17-7 

24-3 

14 

0-024 

0 198 

0-690 

1-62 

3-18 

5-52 

8-75 

13-0 

18 6 

25-4 

15 

0-026 

0-213 

0-725 

1-70 

3-32 

5-80 

9-20 

13-6 

19-5 

26 6 

10 

0-028 

0-223 

0-780 

1-78 

3-49 

6-04 

9-55 

14-2 

20-4 

27-9 

17 

0-029 

0-236 

0-785 

1-89 

3-64 

6-28 

9-95 

15-0 

21-2 

29-1 

18 

0-030 

0-242 

0-815 

1-94 

3-78 

6-52 

10-3 

15 5 

22-0 

30-2 

19 

0 031 

0-250 

0-850 

200 

3-90 

6-80 

10-7 

16-0 

22 8 

31-2 

20 

0-032 

0-258 

0-875 

2-06 

4-02 

7-00 

11-1 

16-5 

23-0 

32-2 

25 

0 038 

0-300 

1-015 

2-40 

414 

8-12 

12-9 

19-2 

24-2 

3o*l 

30 

0042 

0-338 

114 

2-70 

5-30 

918 

146 

21-6 

31 0 

42-4 

35 

0-017 

0-375 

1-26 

300 

5-89 

10-1 

16-2 

24-0 

34-2 

47-1 

40 

0-050 

0-409 

1-39 

3-27 

641 

11-1 

17-6 

26-2 

375 

51-3 

45 

0-056 

0-415 

150 

3-56 

6-95 

120 

19 0 

28-5 

40-5 

556 

50 

0056 

0-474 

1-61 

3-79 

7-44 

12 9 

20-5 

30-3 

43-2 

59-5 

55 

0-062 

0-501 

172 

4-06 

7-95 

13-8 

21-8 

32-5 

46-2 

636 

60 

0-067 

0538 

1-80 

4-30 

841 

14-4 

23-1 

34-4 

49-1 

67 3 

65 

0-071 

0-570 

1-90 

4-56 

8-88 

151 

24-4 

36-5 

518 

71-0 

70 

0-074 

0597 

2-02 

4-77 

9"36 

16-2 

25-5 

38-2 

544 

74-9 

75 

0-078 

0-625 

2-12 

5-00 

9 77 

16-9 

26-8 

40-0 

56-8 

78-0 

80 

0-081 

0-650 

2-20 

5-20 

10-2 

17 6 

28 0 

41-6 

58-1 

81-6 

85 

0-085 

0 680 

2-30 

5-44 

10-6 

18 4 

29-2 

43-5 

62 0 

85-0 

90 

0-088 

0-705 

238 

5 64 

11 0 

191 

30 5 

45-2 

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445 












































Steamship Performance 


461 


Horsepower In Steamship Performance. 


Displace¬ 
ment in 
tons. 

n 

12 1 

CO 

r—i 

Nautical miles or knot 
14 | 15 i 16 

s per h< 

17 

)ur. 

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250 

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1170 

1430 

1740 

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2478 

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3400 

































462 


Steamship Performance. 


Horsepower in Steamship Performance. 


Displace¬ 
ment in 
tons.’ 

1 

2 1 

Na 

3 

utical n 

4 

tiles or 

5 

knots p 
6 

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7 

8 

9 

10 

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11 

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H 

ii 

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H 

H 

II 

H 

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1000 

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150 

225 

318 

439 

1100 

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12-5 

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100 

160 

239 

338 

467 

l?i00 

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620 

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170 

254 

359 

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1300 

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140 

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112 

179 

267 

378 

523 

1100 

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35-0 

68-7 

119 

189 

281 

398 

549 

1500 

0-562 

4-50 

15-5 

360 

71-9 

124 

197 

295 

417 

515 

1000 

0-578 

4-62 

16-2 

37-0 

75-0 

130 

206 

307 

435 

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1700 

03.94 

4-75 

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135. 

215 

320 

453 

6-5 

1800 

0625 

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17-5 

400 

81-2 

140 

224 

332 

470 

649 

1900 

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181 

42-0 

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145 

231 

345 

488 

673 

3000 

0-700 

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18-8 

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150 

239 

356 

504 

696 

3100 

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90-0 

155 

247 

369 

521 

720 

3300 

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5-88 

200 

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160 

255 

380 

537 

741 

3300 

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206 

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262 

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188 

299 

446 

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3900 

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111 

192 

306 

457 

646 

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3000 

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313 

467 

660 

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320 

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218 

347 

518 

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222 

354 

528 

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3700 

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226 

360 

538 

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133 

230 

367 

548 

774 

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135 

234 

373 

558 

787 

1087 

4000 

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138 

238 

380 

567 

801 

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140 

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814 

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142 

246 

392 

586 

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4300 

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145 

250 

398 

595 

840 

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150 

258 

410 

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416 

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100 

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156 

270 

428 

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434 

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277 

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283 

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6000 

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303 

497 

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322 

512 

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330 

526 

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540 

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210 

364 

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599 

879 

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8000 

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220 

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615 

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403 

640 

955 

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122 

238 

411 

653 

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242 

418 

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126 

246 

426 

683 

1008 

1437 

1972 

10000 

205 

16-4 

551 

131 

255 

441 

714 

1044 

1488 

2042 




























Steamship Performance, 


463 


Horsepower in Steamship Performance. 


Nautical miles or knots per hour. 


tons. 

11 

12 

1 O 

lo 

14 

15 

16 

17 

18 

19 

20 

T 

14 

II 

ii 

V 

ii 

H 

H 

ii 

H 

n 

H 

1000 

585 

759 

963 

1206 

1480 

1798 

2157 

2560 

3008 

3514 

1100 

622 

806 

1024 

1284 

1574 

1913 

2295 

2723 

3203 

3736 

1200 

66U 

858 

1090 

1360 

1670 

2030 

2435 

2890 

3400 

3967 

1300 

696 

903 

1147 

1432 

1758 

2136 

2564 

3043 

3576 

4178 

1400 

732 

950 

1204 

1508 

1850 

2248 

2697 

3200 

3762 

4394 

1500 

766 

995 

1264 

1580 

1938 

2355 

2825 

3252 

3943 

4605 

3 000 

800 

1038 

1317 

1648 

2020 

2458 

2948 

3500 

4113 

4803 

1700 

833 

1083 

1374 

1718 

2107 

2561 

3072 

3646 

4286 

5006 

1800 

864 

1123 

1422 

1784 

2188 

2660 

3140 

3785 

4448 

5195 

6.000 

897 

1166 

1479 

1850 

2270 

2760 

3310 

3928 

4615 

5390 

2000 

927 

1205 

1527 

1913 

2345 

2854 

3420 

4060 

4770 

5570 

2100 

958 

1247 

1582 

1979 

2382 

2948 

3535 

4195 

4935 

5762 

2200 

988 

1284 

1628 

2037 

2500 

3038 

3642 

4325 

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5935 

2300 

1017 

1324 

1680 

2102 

2578 

3134 

3755 

4460 

5241 

6120 

2400 

1047 

1360 

1723 

2160 

2646 

3220 

3860 

4580 

5386 

6290 

2500 

1077 

1400 

1777 

2222 

2725 

3313 

3970 

4715 

5542 

6470 

2000 

1102 

1435 

1820 

2280 

2796 

8400 

4075 

4835 

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1473 

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2338 

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4180 

4960 

5832 

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2800 

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1508 

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2395 

2935 

3568 

4280 

5076 

5970 

6970 

2900 

1189 

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1960 

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4385 

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7142 

3000 

1215 

1582 

2000 

2508 

3075 

3740 

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5318 

6255 

7300 

3100 

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1614 

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2565 

3145 

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7470 

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1268 

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2616 

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7622 

3300 

1296 

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2671 

3280 

3985 

4775 

5670 

6666 

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1320 

1717 

2178 

2725 

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4870 

5784 

6784 

7936 

3500 

1347 

1750 

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2779 

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4965 

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6936 

8090 

3000 

1373 

1783 

2264 

2830 

3175 

4222 

5060 

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8250 

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4385 

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4453 

5340 

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7444 

8696 

4000 

1473 

1912 

2427 

3038 

3725 

4530 

5430 

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7580 

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1944 

2468 

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7700 

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4200 

1520 

1975 

2507 

3137 

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4300 

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8195 

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4000 

1614 

2100 

2664 

3333 

4087 

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5960 

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8320 

9710 

4700 

1639 

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3382 

4145 

4970 

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2160 

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6130 

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10120 

5000 

1708 

2220 

2817 

3525 

4321 

5253 

6300 

7475 

8792 

10250 

5250 

1760 

2293 

2909 

3640 

4414 

5426 

6507 

7723 

9U81 

10601 

5500 

1822 

2365 

3000 

3755 

4608 

5600 

6715 

7972 

9370 

10953 

5750 

1876 

2436 

3090 

3868 

4744 

5767 

6917 

8204 

9652 

11269 

0000 

1930 

2507 

3180 

3981 

4880 

5935 

7120 

8436 

9935 

11586 

0250 

1982 

2574 

3261 

4094 

5013 

6096 

7313 

8519 

10203 

11902 

0500 

2035 

2642 

3352 

4207 

5146 

6258 

7505 

8603 

10472 

12218 

0750 

2088 

2710 

3438 

4320 

5281 

6419 

7698 

8986 

10741 

12534 

7 000 

2141 

2778 

3524 

4434 

5417 

6580 

7892 

9370 

11010 

12851 

7250 

2191 

2842 

3606 

4531 

5542 

6733 

8076 

9587 

11265 

13152 

7500 

2241 

2907 

3688 

4629 

5668 

6886 

8260 

9805 

11521 

13453 

7750 

2290 

2971 

3770 

4276 

5794 

7039 

8445 

10022 

11776 

13754 

8000 

2340 

3036 

3852 

4824 

5920 

7192 

8628 

10240 

12032 

14056 

8250 

2488 

3098 

3931 

4923 

6042 

7340 

8806 

10451 

12280 

14345 

8500 

2636 

3161 

4011 

5023 

6164 

7488 

8984 

10662 

12528 

14634 

8750 

2784 

3223 

4095 

5123 

6286 

7637 

9162 

10823 

12776 

14922 

9000 

2933 

3286 

4170 

5222 

6408 

77S5 

9340 

11084 

13024 

15211 

9250 

2879 

3346 

4247 

5343 

6516 

7926 

9512 

11289 

13364 

15493 

9500 

2826 

3407 

4324 

5465 

6645 

8068 

9685 

11494 

13505 

15775 

10000 

2720 

3529 

4478 

5708 

6882 

8351 

10030 

11904 

13987 

16340 





























404 

I 


Stability of Vessels. 


To find tlie Momentum of Stability of a Vessel by Experiments. 



W— weight in tons placed on deck at a distance 
r from the cefttre-line and h feet above the load- 
water line, when the vessel.is in equilibrium; v = 
careen angle, d = depth in feet of the centre of 
gravity of the vessel under meta-centre, Q= mo¬ 
mentum of stability in foot-tons. 

Q = W(r cos. v-\- h sin. v ), 


d 


Q 

T sin. v 


Sin. v — 


_Q_ 

'Id' 


Example. The weight W= 15 tons, the centre of gravity of which is placed at r = 
12 feet from centre on deck and h = 8 feet above the water, which careens the 
vessel to an angle v = 2 °. The displacement T= 4288.8 tons. Required, the mo¬ 
mentum of stability Q, and depth centre of gravity d'l 


Q = 15(12 X cos. 2° + 8 X sin. 2°) = 183.08 foot-tons, 

and d= - —- -= 1.223 feet, the depth of the centre 

4288.8 X 0.0349 

of gravity of the vessel, under meta-centre. 


Momentum of Wind on Sails Careening a Vessel in Sailing. 


i 



Let F denote the force of wind in tons, acting at 
right angle to the vessel on the centre of gravity of 
all the sails, = l feet above the centre of gravity of the 
displacement. Then the momentum of the wind will 
be— Q = Fl = Td sin. v. 

Example. The centre of gravity of all the sails 
being i = 35 feet above the centre of gravity of the 
displacement of a vessel of T— 4288.8 tons. The 
force of wind on all the sails F—'l tons. The depth 
of the centre of gravity of the vessel, under meta- 
centre d = 1.223 feet, as found by experiments. Re¬ 
quired, the momentum Q of the wind, and to what 
angle the vessel will be careened ? 

Q = 7 X 35 = 245 foot-tons, 


and, 


sin. v = 


_s_ 

Td 


245 

4288.8 X 1-223 


0.04671 


= sin. 2° 40 7 40X the careen angle required. 


















































Tonnage Measurement. 


463 


Tonnage of Vessels.—Old. U. S. Measurement. 

T — tonnage of vessel. L = length of the vessel in feet, from the fore part of the 
stem to the after part of the stern-post, measured on the upper deck. B = greatest 
beam in feet, measured above the main-walls, d = depth of the vessel in feet. For 
double-decked vessels, half the beam B is taken as the depth d. For single-decked 
vessels, the depth is taken from the underside of deck plank to the ceiling of the hold. 

Example. L — 186 feet, B = 30 and d — 15, for a double-decked vessel. Re¬ 
quired, the tonnage? 

T= — (L — O.GB) = - ° - X — 5 (180 — 0.6 X 30) = 795.77 tons. 

95 95 

Custom-House New Tonnage Law, May 6, 1864:. 

An Act to regulate the admeasurement of tonnage of ships and vessels of the TJ. S. 

Be it enacted by the Senate and House of Representatives of the United States of 
America in Congress assembled. That every ship or vessel built within the United 
States, or that may be owned by a citizen or citizens thereof, on or after the first 
day of January, eighteen hundred and sixty-five, shall be measured and registered 
in the manner hereinafter provided; also every ship or vessel that is now owned 
by a citizen or citizens of the United States, and shall be remeasured and reregis¬ 
tered upon her arrival after said day at a port of entry in the United States, and 
prior to her departure therefrom, in the same manner as hereinafter described: 
Provided , That any ship or vessel built within the United States after the passage 
of this Act may be measured and registered in the manner herein provided. 

Sec. 2. And be it further enacted , That the register of every vessel shall express 
her length and breadth, together with her depth, and the height under third or 
spar deck, which shall be ascertained in the following manner: The tonnage deck, 
in vessels having three or more decks to the hull, shall be the second deck from 
below ; in all other cases the upper deck of the hull is to be the tonnage deck. The 
length from the forepart of the outer planking, on the side of the stem, to the 
after part of the main sternpost of screw steamers, and to the after part of the rud¬ 
der-post of all other vessels measured on the top of the tonnage deck, shall be ac¬ 
counted the vessel’s length. The breadth of the broadest part on the outside of the 
vessel shall be accounted the vessel’s breadth of beam. A measure from the under 
side of tonnage deck plank, amidships, to the ceiling of the hold (average thick¬ 
ness) shall be accounted the depth of hold. If the vessel has a third deck, then 
the height from the top of the tonnage-deck plank to the under side of the upper- 
deck plank shall be accounted as the height under the spar deck. All measure¬ 
ments to be taken in feet and fractions of feet; and all fractions of feet shall be 
expressed in decimals. 

Sec. 3. And be it further enacted , That the register tonnage of a vessel shall be 
her entire internal cubic capacity in tons of one hundred cubic feet each, to be 
ascertained as follows: Measure the length of the vessel in a straight line along 
the upper side of the tonnage deck, from’the inside of the inner plank (average 
thickness) at the side of the stem to the inside of the plank on the sterntimbers 
(average thickness), deducting from this length what is due to the rake of the bow 
in the thickness of the deck, and what is due to the rake of the stern timber in the 
thickness of the deck, and also what is due to the rake of the stern timber in one- 
third of the round of the beam ; divide the length so taken into the number of 
equal parts required by the following table according to the class in such table to 
which the vessel belongs: 

Table of Classes. 

Class I.—Vessels of which the tonnage length according to the above measure¬ 
ment is fifty feet or under, into six equal parts. 

Class 2.—Vessels of which the tonnage length according to the above measurement 
is above fifty feet, and not exceeding one hundred feet long, into eight equal parts. 

Class 3.—Vessels of which the tonnage length according to the above measure¬ 
ment is above one hundred feet long, and not exceeding one hundred and fifty 
feet long, into ten equal parts. 

Class 4.—Vessels of which the tonnage length according to the above measure¬ 
ment is above one hundred and fifty feet, and not exceeding two hundred feet long, 
into twelve equal parts. 

Class 5.— Vessels of which the tonnage length according to the above measure- 
ment is above two hundred feet, and not exceeding two hundred and fifty feet 
long, into fourteen equal parts. 


30 









466 


Tonnage Measurement. 


Class 6.— Vessels of which the tonnage length according to the above measure¬ 
ment is above two hundred and fifty feet long, into sixteen equal parts. 

Then, the hold being sufficiently cleared to admit of the required depths and 
breadths being properly taken, find the transverse area of such vessel at each point 
of division of the length as follows: 

* Measure the depth at each point of division from a point at a distance of one- 
third of the round of the beam below such deck, or, in case pf a break below a line 
stretched in continuation thereof, to the upper side of the floor timber, the inside 
of the limber strake, after deducting the average thickness of the ceiling, which 
is between the bilgqplanks and limber strake; then, if the depth at the midship 
division of the length do not exceed sixteen feet, divide each depth into four equal 
parts 1 then measure the inside horizontal breadth, at each of the three points of 
division, and also at the upper and lower points of the depth, extending each meas¬ 
urement to the average thickness of that part of the ceiling which is between the 
points of measurement; number these breadths from above (numbering the upper 
breadth one, and so on down to the lowest breadth); multiply the second and 
fourth by four, and the third by two; add these products together, and to the sum 
add the first breadth and the last, or fifth ; multiply the quantity thus obtained by 
one-third of the common interval between the breadths, and the product shall be 
deemed the .transverse area; but if the midship depth exceed sixteen feet, divide 
each depth into six equal parts, instead of four, and measure as before directed, the 
horizontal breadths at the five points of division, also at the upper and lower points 
of the depth ; number them from above as before; multiply the second, fourth and 
sixth by four, and the third and fifth by two; add these products together, and to 
the sum add the first breadth and the last, or seventh; multiply the quantities 
thus obtained by one-third of the common inte[r]val between the breadths, and 
the product shall be deemed the transverse area. 

Having thus ascertained the transverse area at each point of division of the ves¬ 
sel, as required above, proceed to ascertain the register tonnage of the vessel in 
the following manner: 

Number the areas successively one, two, three, etc., number one being at the 
extreme limit of the length at the bow, and the last number at the extreme limit 
of the length at the stern ; then whether the length be divided, according to table, 
into six or sixteen parts, as in classes one and six, or any intermediate number, as 
in classes two, three, four and five, multiply the second and every even-numbered 
area by four, and the third and every odd-numbered area (except the first and 
last) by two; add these products together, and to the sum add the first and last if 
they yield anything; multiply the quantities thus obtained by one-third of the 
common interval between the areas, and the product will be the cubical contents 
of the space under the tonnage deck; divide this product by one hundred, and the 
quotient, being the tonnage under the tonnage deck, shall be deemed to be the 
register tonnage of the vessel, subject to the additions hereinafter mentioned. 

If there be a break, a poop, or any other permanent closed-in space on the upper 
decks, on the spar deck available for cargo or stores, or for the berthing or accom¬ 
modation of passengers or crew, the tonnage of such space shall be ascertained as 
follows: 

Measure the internal mean length of such space in feet, aDd divide it into an 
even number of equal parts, of which the distance asunder shall be most nearly 
equal to those into which the length of the tonnage deck has been divided; meas¬ 
ure at the middle of its height the inside breadths—namely, one at each end and 
at each of the points of division, numbering them successively, one, two, three, etc.; 
then to the sum of the end breadths, add four times the sum of the even-numbered 
breadths and twice the sum of the odd-numbered breadths, except the first and 
last, and multiply the whole sum by one-third of the common interval between 
the breadths; the product will give the mean horizontal area of such space; then 
measure the mean height between the plank of the decks, and multiply by it the 
mean horizontal area; divide the product by one hundred, and the quotient shall 
be deemed to be the tonnage of such space, and shall be added to the tonnage 
under the tonnage decks, ascertained as aforesaid. 

If a vessel lias a third deck, or spar deck, the tonnage of the space between it 
and the tonnage deck shall be ascertained as follows: 

Measure in feet the inside length of the space, at the middle of its height, from 
the plank at the side of the stem to the plank on the timbers at the stern, and 
divide the length into the same number of equal parts into which the length 
of the tonnage deck is divided; measure (also at the middle of its height) the iu- 

* Chapman’s rule, p. 114. 




Tonnage Measurement. 


467 


side breadth of the space at each of the points of division, also the breadth of the 
stem and the breadth at the stern ; number them successively one, two, three and 
so lorth, commencing at the stem; multiply the second and all other even-num¬ 
bered breadths by four, and the third and all the other odd-numbered breadths 
(except the first and last) by two; to the sum of these products add the first and 
last breadths, multiply the whole sum by one-third of the common interval be¬ 
tween the breadths, and the result will give, in superficial feet, the mean horizon¬ 
tal area of such space; measure the mean height between the plank of the two 
decks, and multiply by it the mean horizontal area; and the product will be the 
cubical contents of the space; divide this product by one hundred, and the quo¬ 
tient shall be deemed to be the tonnage of such space, and shall be added to the 
other tonnage of the vessel, ascertained as aforesaid. And if the vessel has more 
than three decks, the tonnage of each space between decks, above the tonnage 
deck, shall be severally ascertained in manner above described and shall be added 
to the tonnage of the vessel, ascertained as aforesaid. 

In ascertaining the tonnage of open vessels the upper edge of the upper strake 
is to form the boundary line of measurement, and the depth shall be taken from an 
athwartship line, extending from edge of said strake at each division of the length. 

The register of a vessel shall express the number of decks, the tonnage under 
the tonnage deck, that of the between decks, above the tonnage deck; also that of 
the poop or other enclosed spaces above the deck, each separately. In every 
registered United States ship or vessel the number denoting the total registered 
tonnage shall be deeply carved or otherwise permanently marked on her main 
beam, and shall be so continued ; and if at ally time it cease to be so continued, such 
vessel shall no longer be recognized as a registered United States vessel. 

Sec. 4. And be it further enacted, That the charge for the measurement of ton¬ 
nage and certifying the same shall not exceed the sum of one dollar and fifty cents 
for each transverse section under the tonnage deck; and the sum of three dollars 
for measuring each between decks above the tonnage deck; and the sum of one 
dollar and fifty cents for each poop, or closed-il space available for cargo or stores, 
or for the bert hing or accommodation of passengers, or officers and crew, above the 
upper or spar deck. 

Sec. 5. And be it further enacted, That the provisions of this act shall not be 
deemed to apply to any vessel not required by law to be registered, or enrolled, or 
licensed, and all acts and parts of acts inconsistent with the provisions of this act 
are hereby repealed. 

English Tonnage Measurement. 

Divide the length of the upper deck between the after part of the stem and the 
fore part of the stern-post into 6 equal parts, and note the foremost, middle and 
aftermost points of division. Measure the depths at these three points in feet and 
tenths of a foot, also the depths from the under side of the upper deck to the ceil¬ 
ing at the limber strake; or in case of a break in the upper deck, from a line 
stretched in continuation of the deck. For the breadths, divide each depth into 5 
equal parts, and measure the inside breadths at the following points, viz.: at .2 
and .8 from the tipper deck of the foremost and aftermost depths, and at .4 and .8 
from the upper deck of the amidship depth. Take the length, at half the amidship 
depth, from the after part of the stem to the fore part of the stern-post. 

'then, to twice the amidship depth, add the foremost and aftermost depths for 
the sum of the depths; and add together the foremost upper and lower breadths, 3 
times the upper breadth with the lower breadth at the midship, and the upper and 
twice the lower breadth at the after division for the sum of the breadths. 

Multiply together the sum of the depths, the sum of the breadths, and the length, 
and divide the product by 3500, which will give the number of tons, or register. 

If the vessel has a poop or half deck, or a break in the upper deck, measure the 
inside mean length, breadth and height of such part thereof as may be included 
within the bulkhead ; multiply these three measurements together, and divide the 
product by 92.4. The quotient will be the number of tons to be added to the re¬ 
sult as above ascertained. 

For Open Vessels. —The depths are to be taken from the upper edge of the upper 
strake. 

For Steam Vessels. —The t mnage due to the engine-room is deducted from the 
total tonnage computed by (he above rule. 

To determine this, measure the inside length of the engino-room from the fore¬ 
mast to the aftermost bulkhead; then multiply this length by the midship depth 
of the vessel, and the product by the inside amidship breadth at .4 of the depth 
from the deck, and divide the final product by 92.4. 







468 


Centripetal Propeller. 


CENTRIPETAL PROPELLER. 


The Centripetal Propeller has, since the year 1851, fought its way through the 
usual obstructions to success, and is now approved and adopted by the most 
advanced engineers in Europe and America. Frcm the course of progress, it ap¬ 
pears that the form of propeller now in use has not been reached through scientific 
investigations, hut through the usual and expensive course of trials aud errors, by 
which it has gradually approached the form represented on Plate XI., and accord¬ 
ing to the present rate of progress that shape will no doubt be reached and finally 
adopted within a few years more. 

The propellers constructed by John Roach for the Pacific Mail Steamship Com¬ 
pany are upon the centripetal principle, a full description of which is.given in a j 
work entitled “Education and Shipbuilding,” published in the year 1866, by H. C. 
Baird, Philadelphia. 

The h'-licoida! or propelling surface in the common propeller is formed by a 
straight generatrix at right angle to ihe axis; whilst in the centripetal propeller 
that surface is formed by a spiral generatrix constructed in an angle w, Foinmla 7. 
In practice this angle can be assumed to be, 

w = 30° for the fore-edge, and 

w'= 45° for the after-edge of the propeller. 

The difference between the angles w and w' makes the pitch expanding from the 
centre to the periphery. 

Having given the spirals a and e, the spirals b, c and d are obtained by dividing 
the angles into four equal parts, as will be understood by the illustration. 

A straight generatrix inclined®o the axis will give the same helicoidal surface 
as that of the curved generatrix at right angles to the axis; but the inclination 
of the straight generatrix must be according to Formula 8. 

The dotted lines fg h i represent a centripetal propeller with straight inclined 
generatrix. Propellers constructed either as the dotted or drawn lines, or between 
the two cases, will produce the same propelling effect in the water. When the 
propeller is constructed between the two cases represented on the drawing, the 
blades will appear curved in both views. 

The length X of tlie propeller should be from 0 2D to 0 25 D, and the pitch from 
1 5.D io 2D. For very sharp vessels constructed for speed, and when the draft of 
water is over one-half the beam, the pitch may be made 2'5 D. 

One quarter of the pitch is set- off on the centreline from 0 to 8, and the helix 
constructed in the ordinary way. The outer edge of the blades should not follow 
the true helix, but be made slightly concave, as shown iu the drawing, which 
makes the pitch expanding in the direction of the axis. 

The mean pitch of the propeller should be calculated by Formula 3, making 
r = 0-772. 


Example 1. The diameter of a propeller is 10 feet 6 inches, and the angle 
W = 58° at the periphery. Required the pitch P= in feet ? 


P = cot. 58° X 3*14 X 10*5 = 20-6 feet. 


Example 2. The propeller on Plate XI. is of dimensions D=15 feet, X = 5 
feet, W= 57° 30', the slip is 38 per cent, or S= 0'38. What power is required to 
drive it 40 revolutions per minute, H=1 

15 3 X40 3 / \ 

H= --{5X0‘38Xcos.57 o 30 , -)-0Tl | =509 horses, nearly. 

480000 V / 


Example 3. A propeller of diameter D — 12 feet, angle W= 64°, and length 
L = 3 feet 6 inches, is to he driven by a steam engine of 450 horses,, the slip 
S = 0‘28. How r many revolutions will it make per minute, n — ? 


n 


78 

12 



_450_ 

(3-5X0-28Xcos.64°+0-ll) 


= 61 revolutions 


per minute. 















PI tit t r XL 




































































Formulas for Propellers. 


469 


Pitch. 

P=7T D cot. W 1 

r = m, -2 

V 

2 7r r 


Vx 2 —L 2 
D 2 L m 


.,3 


- 4 


a 


Angles. 

Areas. 

Cot.W= —, 5 
ttD 

P 2 Lm 

a = -- ,.9 

P 

360 L 

v ~ — - , - 0 

P 

A ~i™{ L+x )’ -- 10 

I) n 2 S 2 n 

w = -, - 7 

• 102-4 

a = -^-(pp^ \'k 2 D 2 +P 2 \ 11 
2-75V \ r 

_ w P n 

Cot.<P — - «, 8 

180 D 

_ 2-5P' 2 

O — > l- 1 

V^ 2 P 2J rP 2 


H= 


D 2 n*( 


Horsepower and Revolutions. 

78 


480000' 


(zScos.TF+O-llY 13 » = — /- ~ , 14 

\ / P \ LScoa.-W+Q-ll 


Horsepower of Friction. 

h = - RLk mn A -f311-7 P 4 + 26*42 P 2 P 2 + pA, - - -15 

59,400,000 P V r 


D = diameter, R = radius, L = length, and P — pitch of the propeller in feet. 
W = angle of the blades to the centre line. 
v = projecting angle of each blade. 

= centripetal angle for the curved generatrix. 

= angle of inclination of the straight generatrix. 

= projecting area of al! the blades. 

= lielicoidal surface of the propelling side of all the blades. 

= lielicoidal surface of one whole convolution. 

= acting area at right angles to the axis. All areas in square feet. 

= length of any helix at radius r, and m — number of blades. 

= length of external helix of the blade. 

= number of revolutions per minute. 

= horsepower required to drive the propeller. 
h = horsepower required for friction in the water. 
k = friction coefficient. See page 448. 


Tlie pitcli of tlie propeller is equal to the tabular number opposite the 
given angle W, multiplied by the diameter. 


w 

Pitch. 

w 

Pitch. 

W 

Pitch. 

W 

Pitch. 

W 

Pitch. 

W 

Pitch, 

30 

5-45 

40 

3-74 

50 

2-63 

60 

1-81 

70 

1-14 

80 

0-55 

31 

5-23 

41 

3-62 

51 

2-54 

61 

1-74 

71 

Ml 

81 

0-50 

32 

5-03 

42 

3-50 

52 

2-45 

62 

1-67 

72 

1-02 

82 

0-44 

33 

4-85 

43 

3-27 

53 

2-37 

63 

1-60 

73 

0-96 

83 

0-37 

34 

4-66 

44 

3-20 

54 

2-28 

64 

1-53 

74 

0-90 

84 

0-33 

35 

4-50 

45 

3-14 

55 

2-20 

65 

1-46 

75 

0-84 

85 

0-27 

36 

4-33 

46 

3-09 

56 

2-12 

66 

1-40 

76 

0-78 

86 

0-22 

37 

4-17 

47 

2-93 

57 

2-04 

67 

1-33 

77 

0-72 

87 

0-16 

38 

4-02 

48 

2-83 

58 

1-96 

68 

1-27 

78 

0-67 

88 

0-11 

39 

3-88 

49 

2-73 

59 

1-89 

69 

1-20 

79 

0-61 

89 

0-06 


















































470 Simple Elements with Old ami New Equivalents (Chemistry), 


Name o» Elements. 


Aluminium . 

Antimony. 

Arsenicum.. 

Barium. 

Bismuth. 

Boron. 

Bromine. 

Cadmium. 

Caesium .. 

Calcium.. 

Carbon . 

Cerium . 

Chlorine. 

Chromium. 

Cobalt. 

Copper. 

Didymium.. 

Erbium., 

Fluorine. 

Gallium. 

Glusium.. 

Gold (Aurujn).. 

Hydrogen. 

Indium. 

Iodine... 

Iridium .. 

Iron (Ferrum).. 

Lanthanum. 

Lead (Plumbum)...'.., 

Lithium. 

Magnesium. 

Manganese. 

Mercury.. 

Molybdenum. 

Nickel.. 

Niobium . 

Nitrogen. 

Osmium.. 

Oxygen ... 

Palladium. 

Phosphorus .. 

Platinum. 

Potassium.. 

Rhodium. 

Rubidium. 

Ruthenium. 

Selenium. 

Silicon.. 


Sodium (Natron) 

Strontium. 

Sulphur. 

Tantalum. 

Tellurium. 

Thallium .,.. 

Thorium. 

Tin (Stannum)... 

Titanium. 

Tungsten (Wolfram) 

Uranium. 

Vanadium. 

Yttrium. 

Zinc. 

Zirconium. 


Sym¬ 

bol. 

Old 

eqvlt. 

New 

eqvlt. 

Sp. gr. 

Al. 

13.7 

27.5 

2.50 

Sb. 

129.0 

122. 

6.70 

As. 

75. 

75. 

5.80 

Ba. 

68.5 

137.2 

4.70 

Bi. 

210.30 

210. 

9.80 

Bo. 

10.9 

11. 

2.00 

Br. 

80. 

80. 

3.187 

Cd. 

56. 

111.6 

8.60 

Cs. 

132.4 

132.15 


Ca. 

20. 

39.9 

1.57 

C. 

6. 

12. 

3.52 

Ce. 

46. 

141.3 

5.5 

Cl. 

35.5 

35.5 

2.44 

Cr. 

26.3 

52.4 

fi.8 

Co. 

29.5 

58.6 

8.9 

Cu. 

31.7 

63.3 

8.9 

D. 

48. 

147? 


E. 


170.6 


F. 

19. 

19.1 

1.31 

Gr. 


69.9 

5.956 

Gl. 

4.7 

9.25 

2.1 

Au. 

196.44 

196.2 

19.34 

H. 

1. 

1. 

0.0692 

In. 

74. 

113.4 

7.2 

I. 

127. 

127. 

4.94 

Ir. 

98.6 

169.7 

18.68 

Fe. 

28. 

55.9 

7.8 

La. 

46. 

92. 

206.4 


Pb. 

103.6 

11.44 

L. 

7. 

7.022 

0.593 

Mg. 

12.16 

24. 

1.7 

Mil. 

27.40 

54.8 

8. 

Hg. 

100. 

200. 

13.59 

M. 

48. 

95.8 

8.6 

Ni. 

29.5 

58.6 

8.8 

Nb. 

48.8 

94. 

,,,,,, 

N. 

14. 

14.044 

0.971 

Os. 

99.4 

198.6 

10. 

0 . 

8. 

16. 

1.1087 

Pd. 

53.2 

106.2 

11.5 

P. 

31. 

31. 

1.83 

Pt. 

98.6 

196.7 

21.5 

K. 

39. 

39.137 

0.855 

Ro. 

52.2 

104.2 

11. 

Rb. 

85.36 

85.2 

1.52 

Ru. 

52.11 

103.5 

8.6 

Se. 

39.7 

78. 

4.8 

Si. 

14. 

28. 

2.49 

Ag. 

108. 

108. 

10.5 

Na. 

23. 

23.043 

0.972 

Sr. 

43.8 

87.2 

2.54 

S. 

16. 

32. 

2. 

Ta. 

68.8 

182. 

10.7 

Te. 

64.5 

128. 

6.6 

Tl. 

204. 

203.6 

11.8 

Th. 

59.5 

233.9 

7.7 

Sn. 

59. 

117.8 

7.3 

Ti. 

25. 

48. 

5.28 

W. 

92. 

184. 

17. 

U. 

60. 

120. 

10.15 

V. 

68.5 

51.2 

5.5 

Y. 

32.2 

89.6 


Zn. 

32.6 

64.9 

7. 

Zr. 

44.8 

90. 

4.15 


Remarks on the Elements. 

Light metal. Like zinc. 

White metal used in types. 
Metal, steel-gray lustre. 

White metal, fuses at red heat. 
Hard brittle reddish metal. 
Combination with potassium. 
Deep red volatile liquid. 

Very soft and ductile metal. 
Two strong blue lines in spectr. 
Light yellow malleable metal. 
Diamond. Graphite. Coal. 
Little known and less used. 

Gas, greenish-yellow color. 
Dark-gray metal, strong affinity 
Reddish-gray, magnetic metal. 
Yellowish-red ductile metal. 
Little known and less used. 
Classed as a metal. 

Found in fluor spar. 
Silver-white metal. 

Its salt has a sweet taste. 
Standard of value. 

Lightest of gases. 

Dark-blue lines in spectrum. 
Metallic bluish solid. 

Hard white metal. 

The most useful metal. 

Little known and less used. 

Soft and malleable metal. 

White metal, burns brilliantly. 
Burns brilliantly. 

Grayish-white metal. 

White liquid metal. 

White brittle metal. 

White, hard, ductile metal. 

Not generally known. 

Gas without color or taste. 
White and brittle metal. 

Gas, supports life and combus’n. 
Hard ductile white metal. 
Translucent solid easily ignited. 
Heaviest of all metals. 

Brittle metal, melts at 130°. 
White, hard metal. 

Metal little known. 

Most infusible of metals. 

A semi-metallic solid. 

Flint, quartz, glass, and clay. 
Metal of standard value. 
Bluish-white and soft metal. 
White metal like barium. 
Brimstone, widely used. 

Little used. 

Lustre of metal like sulphur. 
Green line in spectrum. 

Not used in the arts. 

White and malleable metal. 

Its oxide used for painting. 

An iron-gray metal. 

A steel-white metal. 

A metal little used. 

Found in Sweden in 1843. 

A bluish-white metal. 

In nature as silicate. 

















































































Binary Compounds, with New Equivalents 


471 


Solid. 1 ; and Salts. 

Formulas. 

Commercial Names and Use. 

Aluminium sulphate. 

A1 2 (S0 4 ) 3 . 

Forpreparing salts of aluminium. 

Ammonium chloride. 

nh 4 ci. 

Sal ammoniac, for soldering. 

Arsenious acid... 

As 2 0 3 . 

White arsenic, poisonous. 

Barium oxide.. 

BaO. 

Baryta, a gray powder. 

Barium sulphate. 

BaS0 7 . 

Heavy Spar. Fr. adult, wt. lead. 

Calcium oxide. 

CaO. 

Quick or caustic lime. 

Camphor. 

c 10 h 16 o. 

Used for making celluloid. 

Carbolic acid. 

c 6 h 6 o. 

Used as a disinfectant. 

Carbonate of lime. 

Ca0,C0 2 . 

Common limestone, marble. 

Chloride of lime. 

CaCl 2 0 2 . 

Bleaching powder. 

Chloride of sodium. 

CINa. 

Commou salt. 

Copper sulphate. 

C 11 SO 4 .. 

Blue stone or vitriol. 

Copper pyrites... 

Cu 2 S.Fe 2 S 2 . 

Pyramidal and tetrahedral crys- 

Cuprous oxide. 

Cu 2 0. 

Red oxide of copper. [tals. 

Gold chloride.. 

AuC1 3 . 

Used in photography. 

Gold mercurv. 

Au 2 Hg. 

Gold amalgam. 

Gun-cotton.. 

C 6 H,(N0 2 ) 3 0 5 . 

Used as an explosive. 

Hydrogen sodium carb’te. 

HNaCO v 

Baking powder, artificial yeast. 

Hydrogen potass, carb’te.. 

HKC0 3 . 

Yeast for raising bread. 

Iron, ferric oxide.. 

Fe 2 0 3 . 

Red hematite, iron ore. 

Iron, ferric hydrate. 

Fe 2 H 6 Og. 

Yellow ochre, iron ore. 

Iron, magnetic oxide. 

Fe 3 0 4 . 

Loadstone, iron ore. 

Iron, bisulphate. 

FeS 2 . 

Pyrites, cube crystals. 

Iron, ferric sulphate. 

FeS0 4 + 7II 2 0. 

Green vitriol, copperas. 

Indigo blue. 

c«h 5 no. 

Used in dyeing. 

Lead chromate. 

Pb0,Cr0 3 . 

Chrome-yellow. 

Lead protoxide. 

PbO. 

Litharge, dryer for oils. 

Lead chloride and oxide... 

(PbCl 2 ,7PbO). 

Pigment, Turner’s yellow. 

Lead carbonate. 

PbO,C0 2 . 

White lead, paint. 

Lead sequi-oxide. 

Pb 3 0 4 . 

Minium, red lead. 

Lead sulphate. 

PbS. 

Galena, lead ore. 

Lapis lazuli. 

2AlP0 4 .MgH 2 0 2 . 

Blue precious stone. 

Malachite. 

CuC0 3 .CuH 2 0 2 . 

Green precious stone. 

Manganese binoxide. 

Mu0 2 . 

For making chlorine and oxygen. 

Mercury chloride.. 

HgCl 2 . 

Corrosive sublimate. 

Mercury sulphide. 

HgS. 

Cinnabar, ore of mercury. 

Oxalic acid. 

c 2 h 2 o 4 . 

A powerful poison. 

Paraffin. 

^27^54' 

For making candles. 

Potassium carbonate.... 

k 2 'C0 3 . 

Used for making glass. 

Potassium chlorate. 

kcio 3 . 

For making oxvgen in mediciue. 

Potassium chromate. 

K 2 Cr0 4 . 

Used for bleaching. Calico print- 

Potassium cyanide. 

KCN. 

Used in photography. [ing. 

Potassium hydrogen. 

HKCOo. 

Cream of tartar. 

Potassium nitrate...... 

kno 3 . 

Saltpetre, prismatic crystals. 

Saccharose. 

( -' 12 ^ 22 (J ll- 

Cane-sugar, gum-arabic. 

Silver chloride. 

AgCl. 

Horn-silver, in photography. 

Silver nitrate . 

AgN0 3 . 

Lunar caustic. 

Silver cyanide ...,. 

AgCN. 

Used in electro-plating. 

Sodium biborate. 

Na 9 B 4 .O 7 . 10 H 9 O. 

Borax, used as a flux. 

Sodium nitrate. 

NaN 0 3 . 

Soda saltpetre, cubic crystals. 

Sodium carbonate. 

Na 2 C0 3 . 

Soda, used for making soap. 

Sodium oxide. 

NaO. 

Soda, oxide of natrium. 

Stannous chloride. 

SnCl 2 . 

Tin-salt, used in dyeing. 

Stannic oxide. 

Sn0 9 . 

Tin-stone, cassiterite. 

Starch. 

^ 6 ^ 1005 - 

Used in washing. 

Stearic acid. 


Solid fat, candles. 

Strychnine. 

2 0^*2 4^ 2 ^ 2 * 

Strong poison. 

Sulphate of soda. 

NaO,SO s + 10II 2 O. 

Glauber salt, colorless prisms. 

Sulphate of lime. 

Ca,S0 2 + 2H 2 0. 

Alabaster, gypsum, plaster Paris. 

Tannic acid. 

^27^22^17* 

For tanning leather. 

Zinc chloride. 

ZnCl 2 . 

Fpr preserving timber. 

Zinc sulphate... 

ZnS0 4 . 

White vitriol, used in medicine. 

Equal proportions of different atoms may be formed into different orders 

and make different substances, as cane-sugar and gum-arabic; also, strych- 

nine and quinine, which 

have the same formula. 





































































472 


Binary Compounds, with New Equivalents 


Liquids. 

W ater. 

Alcohol. Ethyl. 

Methyl alcohol.. 

Ether. 

Chloroform. 

Glycerine. 

Nitro-glycerine. 

Oil of turpentine. 

Benzol. 

Nitro-benzol. 

Aniline. 

Carbon bisulphide. 

Nitric acid. 

Sulphuric acid.. 

Hydrochloric acid. 

Nitro-muriatic acid. 

Citric acid... 

Oxalic acid. 

Quinie acid. 

Quinine. 

Gases. 

Atmospheric air. 

Nitrous oxide. 

Nitric oxide. 

Carbonic acid. 

Carbonic oxide... 

Carburetted hydrogen. 

Olefiant gas. 

Cyanogen. 

Ammonia. 

Cyanhydric acid. 

Hydrogen sulphide. 

Sulphurous anhydride. 


Formulas. 

HoO. 

C 2 H 6 0. 

CH 4 0. 

(C 2 H 5 ) 2 0. 

CHC1 3 . 

(C 3 H 5 )H 3 0 3 . 

c 3 h 5 n 3 o 9 . 

Ci 0 H 16 . 

c 6 h 6 . 

C 6 H 5 (N0 2 ). 

c 6 h 7 n. 

cs 9 . 

hn6 3 . 

h 9 so 4 . 

SCI. 

HN0 3 + 2HC1. 

c 6 H,o 7 . 

C 2 H 9 0 4 2H 9 0. 

c 7 -h 12 o 6 -. 

C 2 oS 24 N 2 0 2 . 


n 4 o. 

n 2 o. 

NO. 


co 2 . 

CO. 

ch 4 . 

c 2 ii 4 . 

NC. 

Nil,. 

HCN. 


H 9 S. 

S0 2 . 


Commercial Names or Use. 

The most abundant liquid. 

Spirit of wine, intoxicating. 
Proof spirit. 

Used as a solvent, anaesthetic. 
Used as an anaesthetic. 

Much used in the arts. 

The most powerful explosive. 
Spirit of turpentine. 

Constituent of coal-tar. 

Forms the main portion of ani- 
For aniline colors. [line. 

A solvent for India-rubber 
Aqua-fortis, oxidizing agent. 

Oil of vitriol, much used. 
Muriatic acid. 

Aqua-regia, dissolves gold. 

Juice of lemons. 

A powerful poison. 

From Peruvian bark. 

For chills and fever. 

Not chemically combined. 
Laughing-gas. 

Extinguishes fire. 

Perfectly consumes coal. 
Suffocating, poisonous. 
Marsh-gas, fire-damp. 
Illuminating gas. 

Produces blue color. 

Hartshorn, volatile alkali. 
Prussic acid, poisonous. 

Used as a reagent in laboratory. 
Used for bleaching straw. 


Proportion of Compounds by Weight or Volume. 


Names. 

Ccu'bon. 

a 

Hydrogen. 

H. 

Oxygen. 

O. 

Nitrogen. 

N. 

Olive oil, by weight. 

772 

133 

95 


Spermaceti oil, “ . 

780 

118 

102 


Castor oil “ . 

740 

103 

157 


Linseed oil “ . 

760 

113 

127 

344 


Alcohol, “ . 

527 

129 


.Sugar, “ . 

432 

68 

500 


Atmosp. air. 

230 

770 

“ air by volume. 



210 

790 

Water, fresh, by weight. 


1 

8 

“ “ “ volume. 


2 

1 


India-rubber by weight. 

853 

147 







To Transform Atomic Formulas into Weights. 

Utile. Multiply together the equivalent (equiv.) and the exponent (exp.) 
of each substance, and the product is the proportion in the compound by 
weight. Divide each weight by its specific gravity, gives the proportions by 
bulk or volume. 

Example 1. The chemical formula for common alcohol is C 2 H 6 0. Required 
its proportioned parts by weight in 1000? 

Equiv. Exp. 

Carbon C 2 = 12 X 2 = 24J f 521-76) 

Hydrogen H e = 1X6= 6>X21'74-< 130-44 Vby weight. 

Oxygen 0 = 16 X 1 = 10 j ( 347-84 j 

1000:46 = 21-74 1000'04 





































































Nitro-G lycerjnk. 


473 


NITRO-GLYCERINE, C 3 H 5 N 3 0». 

Nitroglycerine is an oily liquid of the above composition, which is highly ex¬ 
plosive under peculiar circumstances, but can bo set tire to and burned like alcohol 
without explosion. It explodes by concussion or pressure of about 2000 pounds to 
the square inch, or by the corresponding temperature of about 630° Fahr. suddenly 
applied. 

The explosion of nitro-glycerine is instantaneous, like that of electricity pass¬ 
ing between two points, decomposes a small portion of the air and explodes the 
nitrogen by concussion, which makes the electric spark. Thunder and lightning 
are explosions of a kind of nitro-glycerine formed by electricity in the air. 

Small portions of nitro-glycerine, say half an ounce each, placed (any number) 
within a few feet of one another, if one of them is exploded, all the rest will explode 
simul-iixstaiitasieoxisly. Therefore, when a charge is to be exploded, care 
must be taken that no more of it is in the neighborhood. 

It may appear strange that nitro-glycerine can be so dangerous to handle, when 
it requires the enormous pressure of 2000 pounds to the square inch to explode it; 
but the fluid may be squeezed between surfaces of only one 10.000th part of one 
square inch, when the pressure need be only 3 ounces to explode it. 

The charge of nitro-glycerine in blasting is exploded by a percussion cap placed, 
on the end of a fuse and dipped into the liquid. The fuse explodes the fulminant/ 
in the cap, the concussion of which explodes the charge. On account of the action 
of nitro-glycerine being instantaneous, no tamping is required in the blast-hole, ex¬ 
cept water or loose sand, but even that is not necessary. This explosive is there¬ 
fore entirely unfit for use in firearms, which would be blown to pieces without dis¬ 
charge through the muzzle. 

The many and very serious accidents which have happened by unexpected ex¬ 
plosions of nitro-glycerine have caused it to be forbidden transportation on 
railroads and steamboats, for which a new form of the explosive has been invented, 
which consists in mixing sawdust and some other solid substances with nitro-gly¬ 
cerine, to the form of a moist brown powder, of nearly the same specific gravity 
as that of water. 

Dynamite. 

This powder is called dynamite, and is now manufactured by the nitro-glycerine 
inventor, Alfred Nobel, in Hamburg, and also by the Giant Powder Company, in 
San Francisco, California. 

The strength and instantaneous action of dynamite are precisely the same as 
those of nitro-glycerine, but it is much safer to handle, it is said—more so than 
common gunpowder. The dynamite powder is made up into cartridges of different 
sizes to suit the blast-hole, and is exploded by percussion caps like nitro-glycerine, 
and requires no tamping. It has been employed with great success in blasting im¬ 
mense masses of rock in the Andes, Peru. 

The price of dynamite is higher than that of gunpowder per weight, but its ex¬ 
ecution per price is much greater. The blast-holes for dynamite need be only one- 
half the size of those for gunpowder, with equal execution. 

The instantaneous action of dynamite makes it far superior to gunpowder in 
blasting, but it is unfit for use in firearms. 

Any number of cartridges of dynamite placed in a deep blast-hole with tamp- 
ings of sand between, if one of them is exploded, all the rest will explode simulta¬ 
neously. Small cartridges are made for the percussion cap, and called primers, by 
which the principal charge is exploded. Should a charge fail to explode, put in 
a new fuse and primer. 

Blasting under Water. 

For this purpose the cartridges should be made of strong oiled paper and per¬ 
fectly water-tight, to save the dynamite from moisture. The cartridges should also 
be ballasted, so as to sink easy in water, which can be done by placing a lead ball 
in the bottom and pack the dynamite on the top, after which the cartridge is her¬ 
metically sealed with some varnish insoluble in water. The cartridges (any num¬ 
ber) are guided into the blast-hole through a tube, and finally the primer with the 
fuse, by which the whole.charge is exploded. 

Dynamite is insoluble in water, but will not explode if moist with water. It 
freezes to a snowy mass at 40° Fahr., but its explosive quality i* not impaired 
thereby. At 212° the nitrogen evaporates and spoils the powder. 











4~ f 


Cement, Concrete and Mortar. 


CEMENT, CONCRETE AND MORTAR. 

Homan Cement. Parker's anali/sis. 

One part of common clay to 2£ parts of chalk, set very quick. 

Concrete. Eight parts of pebble or pieces of brick about the size of an egg, 
to 4 parts of scrap river-sand, and 1 part of good lime, mixed with water ail'd 
grouted in, makes a good concrete. 

Lime Mortar. One part of river-sand to two parts of powdered lime, mixed 
with fresh water. 

Hydraulic Mortar. One part of pounded brick powder to two parts of pow¬ 
dered lime mixed with fresh water. This mortar must he laid very thick between 
the bricks, and the latter well soaked in water before laid. 

No. 1. Hydraulic Concrete, by Treussart. 

30 parts of hydraulic lime, measured in bulk before slacked. 

30 “ sand. 

20 “ gravel. 

40 “ broken stone, a hard limestone. 

this concrete diminishes one-fifth in volume after manipulation. 

The mortar is made first, and then mixed with gravel and stone. 

No. 2. Another Concrete, by Treussart. 

33 volumes hydraulic lime unslacked. 

4:5 “ Puzzolano (Pozzulano). 

22 “ sand. 

60 “ broken stone and gravel. 

Asphalte Composition for street pavement, by Colonel Emy. 

2| pints (wine measure) of pure mineral pitch. 

11 lbs. of Gaugeac bitumen. 

17 pints of powdered stone-dust, wood-ashes or minion. 


© 

o CO 
0£ ^ 

t~> ft • 

•S3 o 

O' rP 

O U 'f 

^ 

r—“< ri O 

P si** 

.fi » rd 

TH.2 g 
o 

o' 2 <m 

a . 

C ° 


Cements for Cast Iron. 

Two ounces sal-ammoniac, one ounce sulphur and sixteen ounces of borings or 
filings of cast iron, to be mixed well in a mortar and kept dry. When reqiTired 
for use, take one part of this powder to twenty parts of clear iron borings or fil- 
mgs, mixed thoroughly in a mortar, make the mixture into a stiff paste with a 
little water and then it is ready for use. A little fine grindstone sand improves the 

CG1X1 Gilt, 

Or, one ounce of sal-ammoniac to one hundred weight of iron borings No heat 
allowed to it. 

lhe cubic contents of the joint in inches, divided by 5, is the weight of dry bor¬ 
ings in pounds Avoir, required to make cement to fill the joint nearly. 

Cement for Stone and Brick Work. 

Two parts ashes, three of clay and one of sand, mixed with oil, will resist 
weather equal to marble. 

Broxvn Mortar. 

One part Thomaston lime, two of sand and a small quantity of hair. 

Hydraulic Mortar. 

Three parts of lime, four Puzzolano, one smithy ashes, two of sand and four 
parts ot rolled stone or shingle. 


Crushing Weight in Pounds per Square Inch 


m Portland cement, mixed with different proportions of sand, and at different age of 
___ the mixture in months. 


Age in 
months. 

O 

P 

1 

arts of sail 
2 

d to one of 

3 

cement. 

4 

5 

6 

3 

3800 

2490 

1900 

1500 

1200 

950 

780 

6 

5280 

3550 

2750 

2190 

1800 

1500 

1200 

9 

5980 

4450 

3350 

2700 

2280 

1800 

1440 

12 

6160 

5150 

3850 

3010 

2450 

2050 

1600 


About % of this weight should be depended upon in practice. 

Some iron filings in a very weak solution of sal-ammoniac, mixed with Portland 
cement, increases its strength to double or more. 
























Bricks. 


475 


BRICKS. 

Dimensions. 

Common brick, 8 X H X 2£ inches = 85 cubic inches. 
Front brick, 8£ X 4£ X 2£ “ = 92.8 “ “ 

When laid in a wall with cement, it occupies a space of — 

Common brick, 8J X X 2f inches = 102 cubic inches. 
Front brick, 8£ X 4£ X 2£ = 111 “ “ 


Weight and Bulk of* Bricks. 


Tons. 

Pounds. 

Cub. ft. 

by it 
C. brick. 

N umber 
self. 

F. brick. 

of bricks, 
in wall wit 
C. brick. 

ti cement. 

F. brick. 

1 

2240 

22.4 

448 

416.6 

381 

347 

0.04464 

100 

1 

20 

18.6 

17 

15£ 

2.23 

5000 

50.00 

lOOO 

930 

850 

772 

2.4 

5376 

53.76 

1075 

lOOO 

914 

834 

2.62 . 

5872 

58.72 

1130 

1100 

lOOO 

913 

2.88 

6451 

64.51 

1240 

1200 

1100 

lOOO 


One perch of stone is 24.75 cubic feet. 


Acids for Soldering or Tinning. 

TIN. One part of muriatic acid, with as much zinc as it will dissolve, then add 
two parts of water and some sal-ammoniac. 

BRASS and COPPER. One pound of muriatic acid, four ounces of zinc and 
five ounces of sal-ammoniac. 

ZINC. One pound of muriatic acid, two ounces of sal-ammoniac with all the 
zinc it will dissolve, then add three pints of water. 

IRON. One pound of muriatic acid, six ounces sperm tallow and four ounces of 
sal-ammoniac. 

GOLD and SILVER. One pound muriatic acid, eight ounces sperm tallow and 
eight ounces of sal-ammoniac. 

Silvering Metals. 

Ten parts of nitrate of silver, ten parts common salt, thirty parts cream of tar¬ 
tar. Moisten the powder with water when ready to apply. 


Glues. 

Rice glue. Rice flour mixed in cold water and boiled in china or clay pot; stir it 
well during the boiling. This makes an excellent white glue. 

Houseblose glue. Dissolve the houseblose in strong alcohol, and apply it hot on 
the articles to be glued. This makes a very strong glue which is not soluble in 
water or moisture. 


Barrel Measure. 

A barrel of flour weighs 196 pounds. 
A barrel of pork, 200 pounds. 

A barrel of rice, 600 pounds. 

A barrel of powder, 25 pounds. 

A firkin of butter, 56 pounds. 

A tub of butter, 84 pounds. 


14 pounds, ... 1 stone. 

28 pounds, . . .1 quarter. 

4 quarters, ... 1 cwt. 


Busliel Measure. 

The following are sold by weight per 
bushel: 

Wheat, beans and clover-seed, 60 
pounds to the bushel. 

Corn, rye and flax-seed, 56 pounds. 
Buckwheat, 52 pounds. 

Barley, 48 pounds. 

Oats, 35 pounds. 

Bran, 20 pounds. 

Timothy-seed, 45 pounds. 

Coarse salt, 85 pounds. 


Acre. 

A square of 208.75 feet each way is one acre. 
A circle of 235.5 feet in diameter is one acre. 





















476 


Light and Colors. 



LIGHT AND COLORS. 


Light is the sensation transmitted to thereye, and produces the sense of seeing. 
Light is a component part of heat, and a compound imponderable substance whose 
ingredients depend upon the composition of the burning substance; or, burning 
substances can be analyzed by decomposition of its light in a spectrum. 


Decomposition of Light in the Spectrum. 


Colors. 

Maximum ray. 

Combination of Colors. 

Violet. 

Indigo. 

Blue. 

Green. 

Yellow. 

Orange. 

Red. 

Chemical. 

Electrical. 

Light. 

Heat. 

Primary. 

Blue. 

Yellow. 

Blue. 

Red. 

Yellow. 

Red. 

■ 

Secondary. 

Green. . 

Purple. ' 
Orange. 1 

Tertiary. 

Dark 

Green. 

Brown. 


All the colors of the spectrum mixed together make white, which is proved by 
the decomposition of white light, which makes the seven colors. • 


The velocity of light in planetary space is 192500 miles per second. The velocity 
of light through transparent bodies is not known, but probably varies inverse as 
the square root of the specific gravity of the transparent substance. 

Light passes from the sun to the earth, 95000000 miles, in eight minutes, at 
which rate of velocity light can pass around the earth in one-eighth of a second. 

The intensity of light is inverse as the square of the distance from the luminous 
body. 

The standard unit for measuring the intensity of light is assumed to be that 
produced by a sperm candle, “short 6,” burning 120 grains per hour, 

A spermatic candle 0.85 in diameter burns about 1 inch per hour. 


MOTION OF GAS IN PIPES. 

Letters denote , 

Q = cubic feet of gas passed through the gas-pipe per hour. 

L — length in feet, Z> —diameter in inches, of the pipe. 

H = head of water in inches which presses the gas through the pipe. 

$ = specific gravity of the gas, air being 1. 

n — number of candles required for giving the same light as Q cubic feet of gas 
per hour. 


Q = 780D 2 


V 


HD 
SL’ 


Q 


n 


1 /n 

Q 2 


D = 


1 5 j SL Q 2 

14.35V H ' 


Example. At a distance of Z’ = 6450 feet from the gas-work is required Q = 
910 cubic feet of gas per hour. Head of water being H= 1 inch, specific gravity 
s s= 0.5. Required, the diameter of the pipe D = ? 


D 


1 5 10 . 

:.35\ 


5 X 6450 X 940 2 


14.35 


= 5.396 inches. 


Each light in a room consumes about 4 cubic feet of gas per hour, and ordinary 
street-lights 5 cubic feet. 






























Electricity. 


477 


ELECTRICITY. 


Electro-Chemical Order of 


Electro-positive. 

Potassium. 

Sodium. 

Lithium. 

Barium. 

Strontium. 

Calcium. 

Magnesium. 

Aluminium. 

Uranium. 

Manganese. 

Zinc. 

Iron. 

Nickel. 

Cobalt. 

Cadmium. 

Lead. 

Tin. 

Bismuth. 

Copper. 

Silver. 

Mercury. 

Palladium. 

Platinum. 

Gold. 

Hydrogen. 

Silicon. 

Titanium. 

Tellurium. 

Antimony. 

Carbon. 

Boron. 

Tungsten. 

Molybdenum. 

Vanadium. 

Chromium. 

Arsenicum. 

Phosphorus. 

Iodine. 

Bromine. 

Chlorine. 

Fluorine. 

Nitrogen. 

Selenium. 

Sulphur. 

Oxygen. 

Electro-negativ • 



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Order of Compounds. 

Electro-positive. 

Fur. 

Smooth glass. 
Woollen cloth. 
Feathers. 

Wood. 

Pa per. 

Silk. 

Lac. 

Rough glass. 

Sulphur. 

I o! ton. 

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Order of Conducting Power 

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Well-burnt charcoal. 

Plumbago. 

Concentrated aci ds. 

Powdered charcoal. 

Diluted acids. 

Saline solutions. 

Metallic ores. 

Animal fluids. 

Sea water. 

Spring water. 

Bain water. 

Ice above 13° Fahr. 

Snow. • 

Living vegetables. 

Living animals. 

Steam. 

Salts soluble in water. 

Rarefied air. 

Vapor of alcohol. 

Moist earth and stones. 

Powdered glass. 

Flower of sulphur. 

Dry metallic oxides. 

Oils, the heaviest the 
best. 

Ashes. 

•Transparent crystals. 

Ice below 13° Fahr. 

Phosphorus. 

Lime. 

Dry chalk. 

Caoutchouc. 

Camphor. 

Silicious stones. 

Dry marble. 

Porcelain. 

Baked wood. 

Dry gases and air. 

Leather. 

Parchment. 

Dry paper. 

Feathers. 

Hair. 

Wool. 

Dyed silk. 

Bleached silk. 

Raw silk. 

Diamond. 

Mica. 

All vitrifications. 

Glass. 

Jet. 

Wax. 

Sulphur. 

Resins. 

Amber. 

Shellac. 

Gutta-percha, the worst 
conductor of all. 


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478 


Assaying. 


FIRE-ASSAY OF SILVER AND GOLD ORES. 

From actual practice by the author in California and South America. 

A ssay Composition. 

Gold or silver ores, 400 grains. 

Litharge (oxide of lead), 500 “ 

Carbonate of soda, 240 “ 

Borax, 110 “ 

Charcoal, 20 “ 

Total, 1270 “ 

All the ingredients to he well powdered and mixed before placed in the cruci¬ 
ble. Should the ore contain much sulphur, stick a 3-inch nail in the assay. The 
more galena in the ore, the less litharge is required. Smelt the assay, cupel the 
lead and weigh the remaining button of precious metal. 

Should the button be pure silver, multiply the weight in grains by 100, and the 
product is the value of silver in dollars per ton of ore; if pure gold, multiply by 
1500, and the product is the value in dollars per ton of ore. 

When the button contains both gold and silver, the latter metal must be dis¬ 
solved in nitric acid, for which the alloy must contain at least 3 silver to 1 of gold, 
otherwise the acid will not dissolve it. In case the alloyed button does not con¬ 
tain sufficient silver, it is necessary to add what is required, and melt it into one 
button by blowpipe and charcoal. Hammer the button to a thin leaf and boil it 
in nitric acid; when all the silver is dissolved, the pure gold remains solid. Wash 
the gold in clean water, dry and weigh it. 

Suppose the alloyed button to weigh 2.156 grains, and its color being between 
that of gold and silver, so as to suspect too little of the latter metal; then add, 
say, 2 grains of pure silver, and dissolve the button, weigh the remaining gold, 
which, for example, maybe 1.162 grains. Then 2.156 —1.162=1.994 grains of 
silver in the assay. 

Silver, 1.994 -f- 100 = 199.40 dollars per ton. 

Gold, 1.162 -h 1500 = 1743 “ 

Yalue of the ore, = 1942.40 “ “ 

About one per cent, of the precious metal is lost in the cupelling. 

This rule is sufficiently correct for practical purposes. 

North American Standard. 

p f Gold, 387 ounces, 8000 dollars. 

11 e { Silver, 99 ounces, 128 dollars. 

Peruvian Standard. 

p f Gold, 1 ounce, 24.29 pesos = 19.43 soles. 

( Silver, 1 libra, 25.66 pesos = 20.53 soles. 

One peso = 4 francs; one sole = 5 francs. 


Assay Table I.—North and South American Measures. 

The table will answer for any system of assaying weights. 


Percentage 

Value of Metal per ton 

Yalue of Metal per quin- 

Silver per 
cajon. 

of metal in 

of Ore. 

tal of Ore. 

the ore. 

Gold. 

Silver. 

Gold. 

Silver. 

Per et. 

Dollars. 

Dollars. 

Soles. 

Soles. 

Marcs. 

0.1 

602.924 

39.709 

31.090 

2.053 

12 

0.2 

1205.85 

79.418 

62.179 

4.106 

24 

0.3 

1808.77 

119.127 

93.269 

6.158 

36 

0.4 

2411.69 

158.836 

124.359 

8.211 

48 

0.5 

3014.62 

198.545 

155.448 

10.264 

60 

0.6 

3617.54 

238.254 

186.538 

12.317 

72 

0.7 

4220.45 

277.963 

217.627 

14.370 

84 

0.8 

4823.39 

317.672 

248.717 

16.422 

96 

0.9 

5426.31 

357.381 

279 807 

18.473 

108 

1 per cent. 

6029.24 

397.090 

310.896 

20.528 

120 

North American. 

South American. 


L_ 
























Silver and Gold. 


479 


Suppose the assay to be 112 grammes, and the cupelled button weighs 0.657 of a 
gramme of silver, then 0.657 X 1-00 :112 = 0.586 per cent. 


See Table 


See Table 


0.5 = 10.264 

0.08 = 1.642 
0.006 = 0 .012 

0.586 = 11.918 

= 198.545 
31.767 
1006 = 2.382 

232.694 



Soles per quintal. 


Dollars per ton 
of ore. 


Table II. For Gold and Silver. 


Avoir 

Tons. 

Wei 

dupois. 

Pounds. 

gilts. 

Ounces. 

L’roy. 

Grains. 

Va 

in do 
Gold. 

lue 

liars. 

Silver. 

Go 

Cub. ft. 

Bii 

Id. 

Cub. in. 

lk. 

Sih 
Cub. ft. 

r er. 

Cub. in. 

1 

2000 

29166.6 

14 millions. 

602924 

39709 

1.6643 

2875.91 

3.060 

5287.48 

0.0005 

1 

14.5833 

7000 

301.46 

18.854 

- - 

1.43795 

_ _ 

2.64284 

— 

0.06857 

1 

480 

20.6718 

1.2929 

- - 

0.09859 

_ _ 

0.18129 

— 

0.0002 

0.00283 

1 

0.04306 

0.00269 

- - 

0.00020 

- - 

0.00038 

— 

0.00332 

0.04837 

23.2202 

1 

1.000 

- - 

0.0524 

_ _ 

_ _ 

-- 

0.05304 

0.77346 

371.264 

1.0000 

1 

- - 

_ _ 

_ 

0.1401 

0.60085 

1201.7 

17524.8 

8411900 

362267 

- - 

1 

1728 

1.0000 

1728 

- - 

0.69543 

10.1416 

4867.99 

209.645 

- - 

0.0(1058 

1 

0.00058 

1.00000 

0.32679 

653.577 

9531.34 

4575043 

— 

12976.4 

1.0000 

1728 

1 

1728 

— 

0.378227 

5.51581 

2647.59 

— 

7.5095 

0.00058 

1.0000 

0.00058 

1 


Table III. Gold, Silver and -Platinum. 

Weight in grains per square inch of sheet , thickness by Birmingham gauge for those 

metals, and in inches. 


Bir. G. 

Thick. 

Gold. 

Silver. 

Platin. 

Bir. G. 

Thick. 

Gold. 

Silver. 

No. 

inches. 

grains. 

grains. 

grains. 

No. 

inches. 

grains. 

grains. 

1 

0.004 

20.68 

11.52 

25.50 

19 

0.063 

339.5 

184.7 

2 

0.005 

26.93 

14.40 

31.26 

20 

0.069 

371.8 

201.9 

3 

0.006 

32.19 

17.28 

38.00 

21 

0.075 

404.0 

220.0 

4 

0.008 

42.80 

23.52 

50.43 

22 

0.081 

436.1 

237.2 

5 

0.010 

53.85 

29.28 

58.14 

23 

0.087 

468.8 

255.0 

6 

0.012 

64.46 

35.04 

75.45 

24 

0.093 

5O0.0 

272.6 

7 

0.014 

75.42 

40.80 

87.88 

25 

0.099 

533.3 

290.0 

8 

0.016 

86.58 

46.56 

100.1 

26 

0.105 

566.6 

307.8 

9 

0.018 

97.07 

52.00 

113.2 

27 

0.111 

596.1 

325.5 

10 

0.022 

118.9 

64.32 

138.5 

28 

0.117 

630.0 

342 6 

11 

0.025 

134.6 

72.96 

157.7 

29 

0.124 

673.3 

3635 

12 

0.029 

156.3 

84.96 

182.3 

30 

0.130 

701.5 

380.3 

13 

0.033 

178.2 

96.48 

207.4 

31 

0.136 

730.0 

398.4 

14 

0.038 

204.6 

111.3 

239.5 

32 

0.142 

769.5 

416.3 

15 

0.043 

231.5 

125.8 

270.5 

33 

0.148 

798.5 

433.3 

16 

0.048 

258.8 

140.8 

302.6 

34 

0.152 

837.6 

451.6 

17 

0.053 

285.6 

155.3 

323.8 

35 

0.160 

865.7 

470.5 

18 

0.058 

312.6 

170.0 

365.3 

36 

0.166 

894.0 

486.0 


Platin. 

grains. 

397.5 
435.0 
471.8 

509.2 

548.6 
586.0 
625.0 
663.0 

697.3 
735.0 
783.1 
817.0 
855.0 
892.0 
932.0 
970.0 
1007 
1047 


California Rule for Silver and Gold. 

It is an established custom in California to allow one per cent, for base metal in 
all gold and silver bars from the mines. The fineness is always stamped in parts of 
1000; that is, if a gold bar is stamped 900 fine, it is understood to contain— 

900 parts of pure gold, 

90 parts of pure silver, 

10 parts of base metal, 

in 1000 parts of the bar. 












































480 


Gold and Silver. 


To Find the Value of Gold and Silver Bars. 

Example 1. Required, the value of the pure gold in a bar weighing 989 ounces 
and stamped 797 fine ? 

-r, (790 fine = 16.33.07 ) 

From table | 7fine== . 14 . 47 / dollars - 

Required value of the bar, 989 X 16.47.54 = 16294.17 dollars. 

Example 2. A gold bar weighing 866 ounces has been assayed and stamped to 
860 fine. Required, its total value ? 

Metals. Bui. Fine. Ounces, per Ounce. Value. 

Gold, 366 X 860 = 314.76 X 20.67.18 = $6506.65.57. 

Silver, 366 X 130 = 47.58 X 1.27.29 = 61.51.61. 

Base metal, 366 X 10 = 3.66 no value. _ 

Total amount 1000 = 866 Answer, $6568.17.18. 

The last two figures in the columns of Table IV. are decimals of a cent. 

The fineness of gold is also expressed in carats , 24 for pure gold; that is, a piece 
of gold 18 carats fine is 18 X 1000: 24 = 750 fine. 


Table IV.—Valne of Gold and Silver, per onnce Troy, of 

Different Fineness. 


Finen. 
in 1000 . 

Gold. 

Silver. 

Finen. 
in 1000 . 

Gold. 

Silver. 

Fineness 
in 1000 . 

Gold. 

Silver. 


$ 

cts. 

$ 

cts. 


$ 

cts. 

$. 

cts. 


$ 

cts. 

$ 

cts. 

1 

0 

2.07 

0 

00.13 

290 

5 

99.48 

0 

37.49 

650 

13 

43.67 

0 

84.04 

2 

0 

413 

0 

00.26 

300 

6 

20.16 

0 

38.79 

660 

13 

64.34 

0 

85 33 

3 

0 

6.20 

0 

00.39 

310 

6 

40.83 

0 

40.08 

670 

13 

85.01 

0 

86.63 

4 

0 

8.27 

0 

00.52 

320 

6 

61.50 

0 

41.37 

680 

14 

05.68 

0 

87.62 

5 

0 

10.33 

0 

00.65 

330 

6 

82.17 

0 

42.67 

690 

14 

26.36 

0 

89.21 

6 

0 

12.40 

0 

00.77 

340 

7 

02.84 

0 

43.96 

700 

14 

47.03 

0 

90.51 

7 

0 

14.47 

0 

00.90 

350 

7 

23.51 

0 

45.25 

710 

14 

67.70 

0 

91.80 

8 

0 

16.54 

0 

01.03 

360 

7 

44.19 

0 

46.55 

720 

14 

88.37 

0 

93.09 

9 

0 

18.60 

0 

01.16 

370 

7 

64.86 

0 

47.84 

730 

15 

09.04 

0 

94.51 

10 

0 

20.67 

0 

01.29 

380 

7 

85.53 

0 

49.13 

740 

15 

29.72 

0 

95.68 

20 

0 

41.34 

0 

02.59 

390 

8 

06.20 

0 

50.42 

750 

15 

50.39 

0 

96.97 

30 

0 

62.02 

0 

03.88 

400 

8 

26.87 

0 

51.72 

760 

15 

71.06 

0 

98.26 

40 

0 

82.69 

0 

05.17 

410 

8 

47.55 

0 

53.01 

770 

15 

91.73 

0 

99.56 

50 

1 

03.36 

0 

06.46 

420 

8 

68.22 

0 

54.30 

780 

16 

12.40 

1 

00.85 

60 

1 

24.03 

0 

07.76 

430 

8 

88.89 

0 

55.60 

790 

16 

33.07 

1 

02.14 

70 

1 

44.70 

0 

09.05 

440 

9 

09.56- 

0 

56.89 

800 

16 

53.75 

1 

03.40 

80 

1 

65 37 

0 

10.34 

450 

9 

30.23 

0 

58.18 

810 

16 

74,42 

1 

04.73 

90 

1 

86.05 

0 

11.64 

460 

9 

£0.90 

0 

59.47 

820 

16 

95.09 

1 

06.02 

100 

2 

06 72 

0 

12.93 

470 

9 

71.58 

0 

60.77 

830 

17 

15.76 

1 

07.31 

110 

2 

27.39 

0 

14.22 

4S0 

9 

92.25 

0 

62.06 

840 

17 

36.43 

1 

08.61 

120 

2 

48.06 

0 

15.52 

490 

10 

12.92 

0 

63.35 

850 

17 

57.11 

1 

09.90 

130 

2 

68.73 

0 

16.81 

500 

10 

33.59 

0 

64.65 

860 

17 

77.78 

1 

11.19 

140 

2 

89.41 

9 

18.10 

510 

10 

54.26 

0 

65.94 

870 

17 

98.45 

1 

12.48 

150 

3 

10.08 

0 

19.39 

520 

10 

74 94 

0 

67-23 

880 

IS 

19.12 

1 

13.78 

160 

3 

30.75 

0 

20.69 

530 

10 

95.61 

0 

68-53 

890 

18 

39.79 

1 

15.07 

170 

3 

52.42 

0 

21.98 

540 

11 

16.28 

0 

69-82 

900 

18 

60.16 

1 

16 36 

180 

3 

72 09 

0 

23.2” 

550 

11 

36.95 

0 

71-11 

910 

18 

81.14 

1 

17.66 

190 

3 

92.76 

0 

24.57 

560 

11 

57.62 

0 

72-14 

920 

19 

01.81 

1 

18.95 

200 

4 

13.44 

0 

25.86 

570 

11 

78.29 

0 

73-69 

930 

19 

22.48 

1 

20.24 

210 

4 

34.11 

0 

27.15 

580 

11 

98.97 

0 

74,99 

940 

19 

43.15 

1 

21.54 

220 

4 

54.78 

0 

28.44 

590 

12 

19.64 

0 

76-28 

950 

19 

63.82 

1 

22.83 

230 

4 

75.45 

0 

29 74 

600 

12 

40.31 

0 

77.58 

960 

19 

84.50 

1 

24.12 

240 

4 

96.12 

0 

31.03 

610 

12 

60.98 

0 

78.87 

970 

20 

05.17 

1 

25 41 

250 

5 

16.80 

0 

32.32 

620 

12 

81.65 

0 

80.16 

980 

20 

25.84 

1 

26 71 

260 

5 

37.47 

0 

33.62 

630 

13 

02.33 

0 

81.45 

990 

20 

46.51 

1 

28 00 

270 

5 

58.14 

0 

34.91 

640 

13 

23.00 

0 

82.75 

1000 

20 

67.18 

1 

29.29 

280 

5 

78.81 

1 0 

36 20 













































Chemistry. 


481 



To Refine Silver. 


Dissolve the impure silver in nitric acid, add chloride of sodium (salt) sufficient 
to precipitate all the silver in form of chloride; then all the impurities will remain 
in solution. 

Filter, wash and dry the chloride of silver. Fuse in a crucible two weights of 
carbonate of potash, add gradually one weight of chloride of silver, raise the heat, 
and the pure silver will melt and collect on the bottom. 

Tests for Metals in Solution with Aeids. 

The reagents are placed in the liquid, which precipitates the metal in solution. 

REAGENTS. 

PRECIPITATES. 

SOLUTIONS. 

Sulphate of iron, 

Oxalic acid, 

Potash or soda, 

Gold, as brown powder, 1 
Gold in large flakes, 

Gold, yellow, - J 

Gold in 
aqua-regia. 

Potash or soda, 

Plate of copper, 

Muriatic acid, 

Common salt, 

Tincture of nutgall, 

Silver, dark olive, 
Metallic silver, 

White crudy silver, 

White crudy silver, 
Brown silver, 

Silver in 
nitric acid. 

Potash or soda, 
Ferro-prussiate of potash, 
Carbonate of potash. 

Blue cobalt, 

Green “ 

Red “ 

Cobalt in 
nitric acid. 

Pure water, 

Gallic acid, 

Potash or soda, 

White bismuth, 

Greenish yellow, 

White bismuth, 

Bismuth in 
nitric acid. 

Sulphate of soda, 
Sulphuric acid, 

Infusion of nutgall, 

White lead, 

Lead in 
nitric acid. 

Plate of iron or zinc, 
Potash, 

Ammonia, 

Infusion of nutgall, 

Metallic copper, 

Green copper, 

Azure-blue copper, 
Brown copper, _ 

Copper in 
nitric acid. 

Pure water, 

Plate of iron, 

White antimony, 

Black antimony, 

' Antimony in 4 muriatic 
acid, 1 nitric acid. 

Plate of copper, 
u iron, 

Gallic acid, 

Metallic mercury, 

Dark powder, 

Orange yellow, 

Mercury in 

muriatic or nitric acid. 

Infusion of nutgall, 
Ferro-prussiate of potash, 
Ammonia, 

Black iron, 

Blue iron, 

Dark-red iron, 

Iron in 
muriatic acid. 

Acid Test for Strength and Q/iiality of Iron and Steel. 

This is a subject well worthy of attention by workers in iron and steel. The 
sample to be tested is filed smooth, or polished on all sides, and placed in di¬ 
lute nitric or sulphuric acid for 12 to 24 hours; then wash the sample and dry it. 
The action of the acid has revealed the structure of the sample, from which its 
quality can be decided with great precision. 

The best steel presents a frosty appearance; ordinary steel, honeycombed. Iron 
presents a fibrous structure in the direction in which it has been worked; the best 
iron shows the finest fibres. Should the iron be uneven, or made from a pile of dif¬ 
ferent kinds of iron, all are exposed by the action of the acid. Hammered blooms 
show slag and iron ; gray cast iron shows crystals of graphic carbon ; other cast 
• irons sho"w different figures, all with marked characteristics. 

.—i 


31 














182 


Chemistry. 


Iron Pyrites, Sulpliurets. 

There are two kinds of iron pyrites—namely, proto sulphured and bisulphuret, of 
which tiio latter is generally richest in gold. All iron pyrites are slightly mag¬ 
netic, but the gold seems to destroy the magnetism. The protosulphnret acts sen 
sibly on the magnetic needle, whilst the bisulphuret does not, and may therefore 
be distinguished for gold. 

The presence of arseniuret of iron in sulphurets indicates richness in gold. 

• Roasting of Sulplmrets. 

When sulphurets contain magnesia, lime or arsenic, sufficient salt should he 
added to chlorize those substances, which then evaporate and go out through 
the chimney. The amount of those impurities should be ascertained beforehand!. 
The salt should he well mixed with the ore before put into the furnace. Ten 
pounds of salt contain six pounds of chlorine and four pounds of sodium. 


Ten pounds of 

Those impurities are very f Magnesia, . 
injurious to chlorination of d Calcium, 
the gold iii the vat. ^ Arsenic, 


JPotmds required. 


Chlorine. 
3.58 
5.78 
10.64 


Salt. 

6 

9.65 

17.6 


Chlorination of Gold in Roasted Sulplmrets. 

Free gold is attacked and dissolved by chlorine gas, and forms two chlorides, 
namely, 


An. 844 parts of gold. 

Cl. 156 “ chlorine. 

An. Cl. 1000 protochloride of gold. 


An. 648.5 parts of gold. 

Clz. 351.5 " chlorine. 

An. Clz. 1000 terchloride.of gold. 


Gold-hearing sulphurets are roasted for the purpose of obtaining the gold free 
for the action of chlorine gas. The combination is very slow, and requires the gold 
to be very fine for the prompt formation of chloride. In some pres, the gold is too 
coarse for chlorination, when it must be extracted by amalgamation. 

Composition for Making Chlorine Gas. 

For each ton of roasted ore in the vat are required 14 pounds of salt, 10 pounds- 
of peroxide of manganese and 5 quarts of sulphuric acid. The composition should 
be constantly stirred in the gasometer, and kept to a uniform temperature of about 
180° Fahr. The chlorine gas thus formed is led into the vat containing the ore. 

On account of chlorine gas being much heavier than air, the gasometer ought 
to be placed at a considerable height above the vat, to facilitate the chlorination 
of the gold. In California they place the gasometer below the vat, which is de¬ 
cidedly wrong. 

Chloride of gold is soluble in water, and can be washed out from the vat simply 
by pouring water on the top of the ore and running it into another vessel, where 
the gold is precipitated with sulphate of iron. 

Chloride of silver is not soluble in water, and remains in the ore in the vat. 
There is always some silver in gold sulphurets. 

Q,uartz Mills. 

Each stamp, weighing about 800 pounds, lifted one foot 60 times per minute, can 
crush one ton of quartz per 24 hours with a dynamic effect of two horse-power. 
This is the average performance. The custom-mill in Grass Valley, California, 
crushes quartz for about four dollars per ton. 

The stamps are generally divided into sets of four or five, working in one mor¬ 
tar, and called a battery. The shoes and dies in the battery are made o-f chilled 
cast iron. 

Most of the gold is collected by amalgamation in the battery. The pulp from 
the battery contains much gold, which is often allowed to run away, hut generally 
the sulphuret in the pulp is concentrated and roasted for chlorination; the rest of 
the pulp is ground in pans and the gold amalgamated. 

Amalgams. 

GOLD. One weight of mercury amalgamates with two weights of gold. 

SILVER. 10 silver to 19 mercury. 

7 “ “ 20 

TIN. 1 tin to 3 mercury, for looking-glasses. 

1 tin, 1 lead, 2 bismuth, 10 mercury, for glass-globes. 

1 tin. 1 zinc. 3 mercury, for rubbers in electric machines. 










Optics. 


483 
-1 


OPTICS. 


Optics is that branch of philosophy which treats of the property and motion 
of light. 

Mirrors* 

Example 1. Fig. 307. Before a coucave mirror of r = 6 feet radios, is placed 
an object 0 = 1, at d — 1-75 feet from the vertex. Required the size of the 
image 1—1 

T 0 r 1X6 0 . 

image 1 - =——= 2*4 


r —2 d 


6—2X1-75 


• Example 2. Fig. 308. Before a concave mirror of r — 5'25 feet radius, is placed 
an object O — 1, at D = 4 - 5 feet from the vertex. Required the size of the in* 
▼erted image 1—1 

T Or 1X5-25 , . 

intake 1 — ———- = —-- — — 1-4 


2 D—r 


2X4-5—5-25 


Example 3. Fig. 309. Before a convex mirror of r — 1-8 feet radius, is placed 
an object 0 = 1, at D = 3-15 feet from the vertex. Required the size of tho 
image 1—1, and the distauce in the mirror d = 1 

imao-e I— —=0*222 distance d = =0*699 ft. 


2X3*15+1*8 


2X3*15 fl-8 


215 


Example 4. Fig. 310. A. parabolic mirror is h = 1-31 feet high, and d 
feet in diameter. Required the focal distance/ = ? from the vertex. 

focal distance f— - -^1^ =■ 2'646 inches. 

16 h 16X1-31 


Optical Lenses* 

Example 5. Fig. 316. A double convex lens, of crown glass, having its radii 
JR = r — 6 inches. Required its principal focal distance/ = ? 

For crown glass the index of refraction is to = 1’52. See table. 

/= 6 


2(1*52—1) 


= 5*768 inches 


L 


Microscope 

Letters aenobc. 

p = magnifying power of a lens. 

23 = limit of distinct vision. 

a = limit of distinct sight, which for long-sighted eyes is about 10 or 12 
inches, and near-sighted 6 to 8 inches. For common eyes take 
a = 10 inches. 

ft = limit distance of the object from the optical centre at distinct vision. 

Example 6 . Fig. 322. Required the magnifying power of a single microscopo 
-.vith principal focal distance, / = 4‘3 inches 1 

Mag. power p = - - ^ 3- = ^ 3 =3'325 times. 

/ 4-3 

























484 Optics. 



307 

Spherical Concave Mirror. 

r = radius, and f —\r, focal distance of the 
mirror. 

/— 9. r D— dr 

r —2 d ' r —2 d ’ 

The image disappears when d =f = 4 r. 

0 r '^ 

^ Spherical Concave Mirror. 

i- 0r . d- Dr . 

2 D—r 2 D—r 

When the object is beyond the focal 
distance the image will be inverted. 

4 • & 

309 

Spherical Convex Mirror. 

j _ Or Dr 

2 D+r d ~~ 2 D+r 

-!—N 

\ \V 

310 

Parabolic Concave Mirror. 

f d ' ■ n= d '■ 


16 h 11 16/ 


3]— 

311 

Hyperbolic Concave Mirror. 

Heat, Light, or Sound emanating from 
the foci of a hyperbola will be reflected 
diverged, from the concave surface. 

/^P 

pi, - 

312 

Eliptic Concave Mirror. 

Emanating rays from either of the two 
foci in an elipse, will be refracted by the 
convex surface to the other foci. 






































OPTICS, 




Astronomical Telescopes and Opera Glasses* 

Example 7. Fig. 325. The principal focal distance f — 0-65 inches of the 
ocular or eye-lens. F — 58 inches the principal focal distance of the objcctive- 
lens. Required the magnifying power of the telescope 1 — ? 


image I — 


0 F 1X58 qq nn .. ,, , . . 

—-— = - = 89 23 times the obiect. 

/ 0-65 J 


The telescope is to be used at the limited distance D = 1380 feet and D = co. 
Required the proper lengths l — l and micrometrical motion of the ocular or 
eye-lens ? when the limit of distinct sight a = 10in. F = 58 : 12 — 4‘833 feet. 
/ = 0*65 : 12 = 0-05416 feet. 

1380X4-833 10X0-05416 4*89035 

+ 


l = 


1380—4-833 10+0-05416 “04)5386 

When D — 1380 feet, the length l = 4*94421 
When JD = oo, 1 = 4*8333 + 0-05386 = 4*88719 


feet. 


Micrometrical motion of eye lens 


0-05702 „ 
0*68424 inches. 

ll 

15 nearly. 


Table of Refractive Indices* 


Substances. 

Index. 

m. 

Substances. 

Index. 

m. 

Cromate of Lead 

Realgar ... 
Diamond - 
Glass, flint 

Glass, crown 

Oil of Cas-sia 

Oil of Olives 

J 2-97 
t 2-50 

2-55 

2-45 

1-57 

1-52 

1-63 

147 

Quartz- - 

Muriatic Acid 

Water - - - 

Ice .... 

Hydrogen - 
Oxygeu 

Atmospheric air- 

1.54 

1.40 

1.33 

1.30 

1.000138 

1.000272 

1.000294 



314 Pi ism. 

An emergent rays of light a a' falling upon a 
transparent medium A (say a glass prism) will be 
transmitted through in the direction a h, and de ! 
livered in the direction b b', parallel to a a' a". 

V = angle of incident, v = angle of refraction. 

T • e c j.* SX71 . V~ 

lndix oi retraction m =—,-. 

sm. v 


315 Given the direction of the emergent rays a a\ 

augles e and r _to find the angles u and x ,—or 

the direction of the rays b b\ 

cos.z— -, cos. u=m cos A180—z—r). 

m 

x = 180 — (e+r+w). 

When e = m, the angle x is smallest. 

An eye in b’ will see the candle in the direo> 
tioc b' b b". 





































186 Optic?. 


IIBIIlHllllllllll lllilll 

316 Double Convex Lens. | 

r= 4 - R r the principal 

(m—l)(R+r) focal distance. 

/ = ~o7~ —Tv * when R = r 

2 [m —1) 

o .= optical centre of the lens. 

fjj|i 

317 

Plano Convex Lens. 

/- + r . 

J m—l 

The optical centre is in the convex 
surface. 

BB 

Convex-concave Lens [Meniscus.) 

f-+ Rr 

(m —l)(i?—r) 

Draw the radii R' and r' parallel to 
one another.—Draw n o, then o is th® 
optical centre. 


r si9 

Double Concave Lens. 

n R r 

J ~~ (m—l)(R+r)‘ 

111 

Plano Concave Lens. 

f-- r . 

J m—l 

The optical centre is in the concave 

surface. j 

U3 

321 

Concavo-convex Lens . 

^ Rr 

J [m~l)[R-v]' 

Draw R and r' parallel to one an other. 
Draw n o } then o is the optical centre. 

j 






































OPTICS. 


4S7 


* 

li i 

622 

Single Microscope. 

I:0=f:f-*, I -V, D-M, 

»-t* -t*. -#■ 

m 

323 

When the object 0 is beyond the focal 
distance the image I will be inverted. 

I- 0 = f- D f I = ^ ^ d — 

J .U J.V J, i d _p a 


324- Diminishing Power of a Double 
Concave Lens. 

I-.O-f-.f+D 

D=f(°J I \ ,=// 7 . 

325 Astronomical Telescope. 

<f _ Id 


. 4 * 

/- JLf, d=M. 

= l ~ F+ i r+/) 

< i 

I:0-F:f 

DF* + uf 
D—F~ % + J 


* 4* for astronomical telescope, — for opera-glasses. 


326 Opera Glass. 



Formulas are tlie same as for Astronomical Telescope. 
































488 


Geography. 


GEOGRAPHY. 



The Earth on which we live is a round ball or sphere, with a mean diameter of 
7914 statute miles. The whole surface of the earth is 196,800,000 square miles, of 
which only one-fourth or nearly 50,000,000 square miles is land, and about 
150,000,000 square miles water. 


Tatole of Area and Population of the Whole Earth, 1883 . 


Divisions of the Earth. 

America,. 

Europe,. 

Asia, . . . , . . 

Africa,. 

Oceanica,. 

Area in Square 
Miles. 
14.491.000 
3,760,000 
16,313.000 
10,936,000 
4,500,000 

Population. 

100,466,000 

327,743,400 

795,591,000 

205,823,260 

31,619,000 

Proportion to 
Square Mile. 

7 

87 

49 

20 

7 

Total, .... 

50,000,000 

1,461,242,660 

30 


About -pj-th of the whole population are bom every year, and nearly an equal 
number die in the same time; making about one bom and one dead per second. 

The annual increase of population per 1000 is about 6 in Europe and 19 in 
America. Europe loses and America gains by emigration. The annual increase 
of population in the whole world is about 6 per 1000. 

The Earth is not a perfect sphere, it is flatted at the Poles. The following are 
her true dimensions in statute miles of 5280 feet. 

Dimensions of the Earth. 

("7898.8809 miles at the Poles. 

Diameter, . . -< 7911.92 miles mean, or in 45° Iat. 

(7924.911 miles at the Equator. 

Difference, . . 26.0302 miles Poles and Equator. 

Flatted, . . . 13.015 miles at each Pole. 

f24802.486 miles round the Poles. 

Circumference, -< 24851.640 miles mean, or in 45° lat. 

(24884.22 miles round the Equator. 

To Find the Radius of the Earth in Any Given Latitude. 

R = 3955.96(1 -|- 0.00164 cos.2Z), statute miles. 





























Geography, 


4S9 


Distribution of Population in tbe World. 


Br. Amer., 

5,800,000 

Denmark, 

2,700.000 

j Puebla, . . 

75.500 

Montreal, 

102,000 

Copenhagen, 

180,000 

Guanajuato, 

63,000 

Quebec, . . 

62,000 

Russia, . . 

70,000.000 

Cuba, . . 

1,420,000 

Toronto, . . 

45,000 

St. Petersb’rg, 

680.000 

Havana, . . 

209.000 

St. John, N.B. 

37,000 

Moscow, . . 

400,000 

St.Jago Cuba, 

100,000 

Halifax, N. S. 

28,000 

England, 

22,703,000 

Porto Rico 

616,000 

Ottawa, Ont., 

15.000 

Loudon, . . 

Scotland, 

3,883,000 

C. America 

2,665,000 

U. S. Amer., 

38.550,000 

3,359,000 

Whites, 

133,000 

N.York &Brk. 

1,338,600 

Edinburgh,. 

180.000 

Indians. 

1,500,000 

Philadelphia, 

675,000 

Glasgow, . 

468,000 

Negroes, 

30,000 

Chicago, . . 

2!>9,000 

Ireland, . 

6,000,000 

Mixed, 

1.000,000 

St. Louis, . 

311,000 

Dublin, . . 

322,000 

Guatemala 

1.180,000 

Baltimore, . 

268,000 

France, Rp., 

36.595,000 

Guatemala, A 

40,000 

Boston, . . 

251.000 

Paris, . . 

Germ. Emp 

1,830,000 

St Salvador 

600,000 

Cincinnati, . 

216,000 

41,000,000 

St. Salvador,A 

20,000 

S’n Francisco, 

150,000 

Berlin,* . . 

702.000 

Nicai'agua, 

400,000 

Washington, 

120.000 

Austria, 

20,400,000 

Managua, . 

10,000 

Buffalo, . . 

118,000 

Vienna,. . 

607,500 

Honduras, 

350,000 

Newark, . . 

105,000 

Hungary, 

15,509,000 

Comayagua, 

8,000 

Louisville, . 

101,000 

Pesth, . . 

202,000 

Costa Rica, 

135.000 

Cleveland, . 

93,000 

Holland, . 

3,688,500 

San Jose, . 

25,000 

Pittsburg, . 

87,000 

Amsterdam, 

281,800 

S. America, 

31.369,000 

Jersey City, 

82,500 

Bavaria, . 

4,825,000 

Wild Indians, 

3,500,000 
10 000,000 

Detroit. . . 

80,000 

Munich, 

171,000 

Whites, 

Milwaukee,. 

71,500 

Switzerl’d, 

2,670.000 

Negroes, 

600,000 

Providence, 

69,000 

Berne, . . 

29,500 

Mixed, 

14,569.000 

Albany, . . 

69,500 

Belgium, . 

5,022,000 

IT.S.Colom. 

3,000,000 

Rochester, . 

62.500 

Brussels, . 

287,000 

Bogota, . . 

50.000 

Alleghany, . 

53,200 

Spain, . . 

16,642.000 

Panama, . . 

Venezuela, 

20,000 

Richmond, 

51,000 

Madrid, . . 

317,000 

1,565,000 

New Haven, 

51,U00 

Italy,. . . 

26,000.000 

Caraccas, 

48,000 

Charleston, . 

49,000 

Rome, . . 

240,000 

Equador, 

1,110,000 

Troy, . . 

46.500 

Greece, . . 

1,458,000 

Quito,. . . 

76.000 

Syracuse, 

43.000 

Athens,. . 

42,000 

Guiana, . 

221,000 

Indianapolis, 

41,500 

Turkey,E., 

15,487.000 

Georgetown, 

26,000 

Worcester, . 

41,100 

Const’tinople, 

1,075.000 

Brazil, . . 

11,780,000 

Lowell, . . 

41,000 

Turkey, A., 

16,463.000 

Rio Janeiro, 

420.000 

Memphis, 

40,200 

Smyrna, 

150.000 

Bahia, . . 
Slaves, 

152,000 

Cambridge, 

40,000 

Arabia, . . 

8,000,000 

1,400,000 

Hartford, 

37.200 

Mecca, . . 

60,000 

Peru, . . 

3,100.000 

Scranton, . 

35,100 

Persia, . . 

9,000,000 

Lima, . . . 

130 000 

Reading, . . 

34,000 

Tabreez, 

110,000 

Callao, . . 

40,000 

Kansas City, 

32,300 

Afgb’nist’n 

4,000.000 

Bolivia, 

1,990,000 

Mobile, . . 

32,000 

Candabar, . 

100,000 

La Paz, . . 

22,000 

Portland, . 

31,500 

Beloocbs’n 

1,500 000 

CbiJi, . . 

1,909,000 

Wilmington, 

31,000 

Kelat, . . 

15,000 

Santiago, . 

115,380 

Toledo, . . 

31,600 

Turkistan, 

6,500.000 

100.000 

Valparaiso, . 

70.500 

Columbus, . 

31,300 

Bokhara, . 

Argentine, 

1.750.000 

Dayton,. . 

30,500 

India, . . 

172,000,000 

Buenos Ayres 

177,800 

Lawrence, . 

29,000 

Bombay, . 

817,000 

Paraguay, 

1.330,000 

Utica, . . 

28,800 

Calcutta,. . 

616,000 

Asuncion, . 

15,000 

Savannah, . 

28,300 

China,. . 

446,500,000 

Uruguay, 

2,614,800 

Nashville, . 

26,000 

Peking, . . 

1,800,000 

Montevideo, 

Patagonia, 

128.000 

Alaska, . . 

54,000 

Canton,. . 

1,000,000 

1,200,000 

Sitka, . . 

? 

Hong-Kong, 

40.000 

Antonio, . . 

? 

Races in U. S. 
White, 

33,570,000 

•Japan, 

Yeddo, . . 

35.000,000 

2,000.000 

Australia, 

Melbourne, . 

1,900,000 

193.700 

Colored, 

4,890.000 

Miaco, . . 

500,000 

Wellington, 

3.300 

Chinese, 

64,000 

Barbary, . 

2,800,000 

Jamaica, 

444,000 

Indians, 

25,500 

Tunis, . . 

130.000 

Kingston, . 

36,500 

S werlen, . 

4.106,000 

Egypt, . . 

Cairo, . . 

5,195,000 

313.400 

Hayti, . . 

P’t-au-Prince 

572.000 
21 000 

Stockholm, 

136,000 

Jerusalem. . 

25,000 

Sandwich I 

73.000 

Norway, . 

1,650,000 

Mexico, . 

9,200.000 

Honolulu, . 

13 530 

Christiana, 

63,500 

Mexico City, 

200,000 s | W. Indies, 

4,000,000; 

- 1 






















Latitude and Longitude. 


49* 


Latitude and Longitude of Places (from Greenwich.) 


America 

Latitude. 

Longitude, j 


Latitude. 

Longitude. 

Atl. Coast. 

D. M. S. 


D M. S. 


Prance. 

D. M. S. 


D. M. S. 


Quebec . . . 

46.49. 

N 

71.16. 

W. 

Paris, Obs. . . 

48.59.13 

N. 

0.09.21 E. 

Halifax . . . 

44.38. 

a 

63.35. 

66 

Cherbourg . . 

49.38. 


1.37. 

66 

Chicago . . . 

42.00. 

u 

87.35. 

66 

Marseilles. 

43.18. 


5.22. 

66 

Boston . . . 

42.21. 

a 

71.04. 

66 

Calais . . . 

50.58. 


1.51. 

66 

New York . . 

40.42. 

a 

74.00.42 

66 

Brussels. . . 

50.51. 


4.22. 

6i 

Philadelphia . 

39.57. 

66 

75.10. 

<( 

Antwerp . . 

51.13. 


4*24. 

6 . 

Cincinnati . . 

39.06. 

u 

84.30. 


Italy. 





St. Louis . . 

38.36. 

u 

89.36. 

66 

Turin . . . 

45.04.06 


7.42. 

U 

Washington . 

38.53. 

a 

77.00.18 

66 

Florence . . 

43.46. 


11.16. 

66 

Ch-ulsston. . 

32.42. 

u 

79.54. 

66 

Leghorn . . 

43.32. 


10.18. 

*• 

New Orleans . 

29.57.30 

u 

90.00. 

66 

Rome . . . 

41.54. 


12.27. 


Georgetown,Br. 

32.22.12 

a 

46.37.06 

66 

Malta . . . 

35.54. 


14.30. 

66 

Nassau . . . 

25.05.12 

u 

77.21.12 


Naples . . . 

40.50. 


14.16. 


Port-au-Prince 

19.46.24 

66 

72.11.12 

66 

Palermo . . 

38.08. 


13.22. 

66 

Porto Rico. . 

18.29. 

66 

66.07.06 

66 

Venice . . . 

45.26. 


12.21. 

66 

Kingston, Jam. 

17.58. 

66 

76.46. 

66 

Austria. 





Havana . . . 

23.09. 

66 

82.22. 

66 

Vienna . . . 

48.13. 


16.23. 

66 

Vera Cruz . . 

19.12. 

66 

96.09. 

66 

Trieste . . . 

45.39. 


13.46. 

66 

Mexico, City . 

19.26. 

66 

99.05. 

66 

Pesth.... 

47.28. 


19.13. 

66 

Colon, N. G. . 

9.22. 

66 

79.55. 

66 

Germany. 





Para .... 

1.28. 

S. 

4S.29. 

66 

Berlin . . . 

52.31. 


13.24. 

<t 

Rio Janeiro . 

22.56. 

66 

43.09. 

66 

Hamburg . . 

53.33. 

tC 

9.56. 

66 

Buenos Ayres 

34.36. 

66 

58.22. 

66 

Cologne . . 

50.56. 


6.58. 

66 

Cape Horn . . 

55.59. 

66 

67.16. 

66 

Amsterdam . 

52.22. 

66 

4.51. 

66 

Pac. Coast. 





Bremen . . . 

53.05. 


8.49. 

6% 

Valparaiso . . 

33.02. 

66 

71.41. 

66 

Berne . . . 

46.57. 


7.25. 

66 

Callao . . . 

12.04. 

6i 

79.13. 


Turkey. 





Lima* . . . 

12.02.34 

66 

79.06. 

66 

Constantinople 

41.01. 

66 

28.59. 

66 

Cuzco* . . . 

13.31.45 

66 

74.15.50 

66 

Ragusa . . . 

42.38. 

66 

18.07. 

66 

Payta . . . 

5.05. 

66 

81.10. 

66 

Salonica . . 

40.39. 

66 

22.57. 

66 

Guayaquil . . 

2.13. 

66 

79.53. 

66 

Athens . . . 

37.5 S. 

66 

23.44. 

U 

Panama. . . 

8.57. 

N. 

79.31. 

66 

Smyrna . . . 

38.26. 

66 

27.07. 

66 

Acapulco . . 

16.55. 

66 

99.48. 

6( 

Cairo .... 

30.03. 

66 

31.18. 

66 

San Francisco 

37.47. 

66 

122.21. 

66 

Jerusalem, Pal. 

31.48. 

66 

37.20. 

66 

Alaska . . . 

58. 


158 


Russia. 





Behring’s Strait 

67°. 


170 


St, Petersburg 

59.56. 

66 

30.19. 

66 

China. Ind. 





Moscow. . . 

55.46. 

66 

35.33. 

66 

Peking . . . 

39.54. 

66 

116.28. 

E. 

Nish Novgorod 

56.20. 

<( 

43.43. 

66 

Canton . . . 

23.07. 

66 

113.14. 

66 

Cazan . . . 

55.48. 

66 

48.50. 

66 

Hongkong. . 

22.15. 

66 

114.12. 

66 

Archangel . . 

64.32. 

66 

40.14. 

66 

Honolulu . . 

21.19. 

66 

157.52. 

66 

Jecatheriuburg 

56.50. 

66 

60.21. 


Jeddo . . . 

35.40. 

66 

139.43. 

66 

Astracan . . 

46.21. 

66 

47.46. 

«( 

Owyhee, S. Isl. 

20.23. 

66 

155.54. 

W 

Odessa . . . 

46.27. 

66 

30.42. 

66 

Calcutta . . 

22.34. 

66 

88.20. 

E. 

Warsaw. . . 

52.13.05 

66 

21.02.9 

66 

Batavia . . . 

6.08. 

66 

106.50. 

66 

Sweden. 





Sydney . . . 

34.00. 

S. 

151.23. 

66 

Stockholm. . 

59.21. 

66 

18.04. 

6k 

Melbourne . . 

37.4S.36 

66 

144.57.45 

66 

Gothenburg . 

57.42. 

66 

11.57. 

a 

Wellington 

41.14. 

66 

174.44. 


Wisby, Gotland 

57.39. 

66 

1S.17. 

u 

Africa. 





Christiania 

59.55. 

66 

10 52. 

66 

Cp. of G. Hope. 

34.22. 

66 

18.30. 

66 

Bergen . . . 

60.24. 

66 

5.20. 

66 

Morocco . . 

39.34. 

N. 

2.23. 

66 

Ystad . . . 

55.25. 

66 

13.50. 

66 

Algiers . . . 

36.47. 

66 

3.04. 

66 

Haparanda 

65.49. 

66 

24.11. 

66 

England. 





Copenhagen . 

55.41. 

66 

12.34. 


London, Tower 

51.31. 

66 

0.06. 

W. 

Spain. 





Greenwich 

51.28.38 

a 

0 0 0 


Madrid . . . 

40.25. 

66 

3.42. 

W. 

Liverpool . . 

53.22. 

66 

2.52. 

66 

Barcelona . . 

41.23. 

66 

2.11. 

E. 

Glasgow . . 

55.52. 

66 

4.16. 

66 

Gibraltar . . 

36.06. 

66 

5.20. 

IV. 

Edinburgh . . 

55.57. 

66 

3.12. 

66 

Carthagena . 

37.36. 

(6 

1.01. 

66 

Bristol . . . 

51.27. 

66 

2.35. 

66 

Lisbon . . . 

38.42. 

66 

9.09. 

66 

Dover . . . 

51.08. 

66 

1.19. 

E. 

Oporto . . . 

41.11. 

66 

8.38. 

66 

Dublin . . . 

53.23. 

66 

6.20 

wj 

Terra, Island 

27.47. 

66 

17.56. 

66 


* Measured by the author. 



















Difference of Longitude in Time. 491 


Difference of Longitude in Time Between Places. 





San 





London. 

St. 

Peters- 

Canton, 




Francisco. 

New York. 

Greenwich. 

burg. 

China. 




H. 

M. 

S. 

H. 

M. 

S. 

H 

M. S. 

%. 

M. S. 

H. 

M. 

S. 

Amsterdam . . . 

• 


8 

29 

19 

5 

15 

32 

0 

19 32 

1 

41 44 

7 

13 

24 

Antwerp . . . 

• 

• 

8 

27 

17 

5 

13 

30 

0 

17 36 

1 

43 40 

7 

15 

20 

Batavia .... 



8 

42 

50 

11 

56 

37 

7 

07 20 

5 

6 4 

0 

25 

36 

Berlin .... 



9 

3 

22 

5 

49 

35 

0 

53 35 

1 

7 41 

6 

39 

21 

Boston .... 



3 

25 

33 

0 

11 

46 

4 

44 14 

6 

45 30 

11 

42 

50 

Buenos Ayres 



4 

16 

19 

1 

2 

32 

3 

53 28 

5 

54 44 

11 

26 

24 

Canton .... 



8 

17 

17 

11 

31 

4 

7 

32 56 

5 

31 40 

0 

0 

0 

Calcutta . . . 



9 

50 

53 

10 

49 

20 

5 

53 20 

3 

52 4 

1 

39 

36 

Cairo. 



10 

i4 

59 

7 

1 

12 

2 

5 12 

0 

■ 3 56 

5 

27 

44 

Copenhagen . . 

. 

# 

9 

0 

3 

5 

46 

16 

0 

50 16 

1 

11 0 

6 

42 

40 

Constantinople 



10 

5 

43 

6 

51 

56 

0 

55 56 

0 

5 20 

5 

37 

0 

Dublin .... 



7 

41 

25 

4 

30 

38 

0 

25 22 

2 

26 38 

8 

18 

18 

Florence . . . . 



8 

54 

51 

5 

41 

4 

0 

45 4 

1 

16 12 

6 

47 

52 

Gibraltar . . . 



7 

47 

56 

4 

34 

32 

0 

21 28 

2 

22 44 

7 

54 

24 

Gothenburg. . . 



8 

57 

38 

5 

43 

51 

0 

47 48 

1 

13 28 

6 

45 

08 

Halifax .... 



3 

54 

27 

0 

41 

40 

4 

14 20 

6 

15 36 

11 

47 

16 

Hamburg. . . . 



8 

49 

39 

5 

35 

52 

0 

39 52 

1 

21 24 

6 

53 

04 

Jecatlierinburg . 

• 


11 

48 

57 

7 

25 

20 

4 

1 16 

2 

0 0 

3 

31 

40 

Jerusalem . . . 



10 

39 

7 

7 

25 

20 

2 

23 20 

0 

22 4 

5 

09 

36 

Lima. 



1 

43 

2 

0 

12 

24 

5 

08 24 

7 

9 40 

11 

18 

40 

London . . . . 



8 

9 

50 

4 

56 

3 

0 

0 24 

2 

1 40 

7 

33 

20 

Lisbon .... 



7 

33 

11 

4 

19 

24 

0 

36 36 

2 

37 52 

8 

08 

32 

Madrid .... 



7 

55 

39 

4 

41 

52 

0 

14 8 

2 

15 24 

7 

47 

04 

Melbourne . . . 



4 

2 

24 

7 

16 

11 

9 

39 51 

11 

41 31 

2 

7 

0 

Naples .... 



9 

6 

51 

5 

53 

4 

0 

57 4 

1 

04 12 

6 

35 

52 

New Orleans . . 



2 

9 

37 

1 

04 

10 

6 

0 10 

8 

01 26 

10 

26 

54 

New York . . . 



3 

13 

47 

0 

0 

0 

4 

56 3 

6 

57 19 

11 

31 

01 

Paris. 



8 

19 

7 

5 

5 

20 

0 

9 20 

1 

51 50 

7 

23 

36 

Peking .... 



8 

4 

21 

12 

41 

52 

7 

45 52 

5 

44 36 

0 

12 

56 

Quebec . . . . 



3 

2 

36 

0 

11 

11 

4 

44 49 

6 

46 5 

11 

19 

53 

Rome .... 



8 

59 

35 

6 

45 

48 

0 

49 48 

1 

11 28 

6 

43 

08 

San Francisco . . 



0 

0 

0 

3 

13 

47 

8 

9 47 

10 

11 3 

8 

17 

17 

St. Louis . . . 



2 

8 

46 

1 

5 

1 

6 

1 1 

8 

2 17 

10 

26 

03 

St. Petersburg . . 



10 

11 

3 

6 

57 

16 

2 

1 16 

0 

0 0 

5 

31 

40 

Stockholm . . . 



9 

22 

11 

6 

8 

24 

1 

12 24 

0 

58 52 

6 

20 

32 

Turin. 



8 

40 

38 

6 

26 

51 

0 

30 48 

l 

30 28 

7 

02 

08 

Washington . . 



3 

1 

46 

0 

12 

1 

5 

8 1 

7 

9 17 

11 

19 

53 

Wellington, N. Z.. 



5 

30 

22 

8 

44 

09 

ii 

38 56 

10 

19 48 

2 

46 

52 


To Deduce Longitude in Degrees into Time, and vice versa. 

Rule 1. Divide the number of degrees by 15, and the quotient is the corre¬ 
sponding hours. Should the degrees be less than 15, multiply them by 4, and the 
product will be minutes in time. The minutes of degrees multiplied by 4 will be 
seconds in time. The seconds of degrees divided by 15 will be seconds in time. 

Rule 2. The time in hours, minutes and seconds, multiplied by 15, will be the 
corresponding angle in degrees, minutes and seconds. The trigonometrical table 
contains the conversion of time and angle. 

Example 1. Required, the difference in time between Philadelphia and Paris ? 

Longitude of Philadelphia, 75° 10' W. 

“ “ Paris, . . 2 20 E. 

Difference in longitude 77° 30' divided by 15 will be 
5/i 10m, the difference in time. When it is 12 o’clock in Philadelphia, it is bli 10 m 
o’clock in Paris. 

Example 2. A vessel sails from New York to Liverpool ; after she has been at 
sea about one week, her difference in time with New York is found to be 2/t7m45s. 
Required, her longitude from New York? 

15(2/4 7 45) = 31° 58' 15" from New York. 

The time is ahead in the east, from where the sun rises. The time is behind in j 
the west, toward sunset. j 




















































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493 


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Astronomy. 


ASTRONOMY. 

The matter constituting the heavenly bodies has probably been evenly distrib¬ 
uted in space from the beginning; the force of attraction gradually formed bodies, 
which accumulated into groups or nebulae, each of which finally became a planet¬ 
ary system with the largest body in the centre, now seen as stars. This operation 
of forming or creating bodies and planetary systems is still and will for ever be 
continued in parts of the infinite space. Each star has a planetary system, and 
astronomers have even been able to observe some planets of the star Sirius. 

The magnitude of this operation, with the enormous dimensions, even within a 
single group, can hardly be conceived by any human mind; for the long row of 
figures expressing a number of conceivable units of length or weight does not 
bring the real magnitude within conception. 

Tlie Sun. 

The sun is a dark body surrounded by a luminous substance in which spots are 
frequently seen. The spots are caused by meteors or other heavenly bodies falling 
into it. 

Mean distance from the earth to the sun, 95,000,000 miles, or 11,992 diameters of 
the earth. 

Inclination of the ecliptic to the equinoctial, 23° 28' 40". 

The sun subtends an angle from the earth of 32' 3". 

Horizontal parallax of the sun, 8.6". 

The Moon. 

Distance from the Earth to the Moon, 273,000 miles, = 30 diameters of Earth, 
or about 0.25 diameters of the Sun, or the diameter of the Sun is twice the diam¬ 
eter of the Moon’s orbit around the Earth. 

Diameter of the Moon, 2160 miles, or about 0.2729 of the diameter of the 
Earth. 

Volume of the Moon, 0.02024 of that of the Earth. 

Density of the Moon, 0.5657 of that of the Earth. 

Mass of the Moon, 0.0114 of that of the Earth. . 

Inclination of the Moon’s orbit to the ecliptic, 5° 8' 48". 

The Moon subtends an angle from the Earth of 31' 7". 

Mean sidereal revolution of the Moon, 27.32166 solar days. 

Mean synodical revolution of the Moon, 29.5305887 solar days. 

The Moon passes the meridian in periods of 24.814 hours, or 48m. 50s. later every 
day. 

Moon’s Age is the number of days from the last new moon. 

! Epact of tlie Year is the Moon’s age on the 1st of January of each year. 
See Almanac for the 19th Century. 

The sum of the epact of the year and that of the month is the moon’s age on 
the first of the month. 


Epact of tlie Month. 


Jan. 

Feb. 

March 

April. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

0 

2 

1 

2 

3 

4 

5 

6 

8 

8 

10 


To Find tlie Moon’s Age on Any Given Day. 

Add together the epacts of the year and month and the date of the month; the 
sum will be the moon’s age. If it exceeds 30, reject that much, and the remainder 
is the moon’s age. 

























Astkonomy. 


497 


Almanac for the 19tli Century. 




Dom. 

•4-3 



Dom. 

4-3 




Dom. 


Yrs. 

Days. 

let- 


Yrs. 

Days. 

let- 



Yrs. 

Days. 

let- 




ter. 




ter. 

S 




ter. 

w 

1800 

Saturd.* 

FE 

4 

1834 

Saturd. 

E 

20 


1868 

Sunday* 

ED 

6, 

1801 

Sunday. 

D 

15 

1835 

Sunday. 

D 

. 1 


1869 

Monday. 

C 

17 

1802 

Monday. 

C 

26 

1836 

Tuesd* 

CB 

12 


1870 

Tuesday. 

B 

28 

1S03 

Tuesday. 

B 

7 

1837 

Wednes. 

A 

23 


1871 

Wednes. 

A 

9 

1804 

Thumb* 

AG 

18 

1838 

Thursd. 

G 

4 


1872 

Friday.* 

GF 

20 

1805 

Friday. 

F 

29 

1839 

Friday. 

F 

15 


1873 

Saturd. 

E 

1 

1806 

Saturd. 

E 

11 

1840 

Sunday* 

ED 

26 


1874 

Sunday. 

D 

12 

1807 

Sunday. 

D 

22 

1841 

Monday. 

C 

7 


1875 

Monday. 

C 

23 

1808 

Tuesd.* 

CB 

3 

1842 

Tuesday. 

P 

18 


1876 

Wedns.* 

BA 

4 

1809 

Wednes. 

A 

14 

1843 

Wednes. 

A 

29 


1877 

Thursd. 

G 

15 

1810 

Thumb 

G 

25 

1844 

Friday.* 

GF 

11 


1878 

Friday. 

F 

26 

1811 

Friday. 

F 

6 

1845 

Saturd. 

E 

22 


-1879 

Saturd. 

E 

7 

1812 

Sunday.* 

ED 

17 

1846 

Sunday. 

D 

3 


1880 

Monday* 

DC 

18 

1813 

Monday. 

C 

28 

1817 

Monday. 

C 

14 


1881 

Tuesday. 

B 

29 

1814 

Tuesday. 

B 

9 

1848 

Wedns.* 

BA 

25 


1882 

Wednes. 

A 

11 

1815 

Wednes. 

A 

20 

1849 

Thursd. 

G 

6 


1883 

Thursd. 

G 

22 

1816 

Friday.* 

GF 

1 

1850 

Friday. 

F 

17 


1884 

Satu rd.* 

FE 


1817 

Saturd. 

E 

12 

1851 

Saturd. 

E 

28 


18S5 

Sunday. 

D 

14 

1818 

Sunday. 

D 

23 

1852 

Mond.* 

DC 

9 


1886 

Monday. 

C 

25 

1819 

Monday. 

€ 

4 

1853 

Tuesday. 

B 

20 


1887 

Tuesday. 

B 

6 

1820 

Wed ns.* 

BA 

15 

1854 

Wednes. 

A 

1 


1888 

Thursd* 

AG 

17 

1821 

Thumb 

G 

26 

1855 

Thursd. 

G 

12 


1889 

Friday. 

F 

28 

1822 

Friday. 

F 

7 

1856 

Saturd.* 

FE 

23 


1890 

Saturd. 

E 

9 

1823 

Saturd. 

E 

18 

1857 

Sunday. 

D 

4 


1891 

Sunday. 

D 

2() 

1824 

Monda * 

DO 

29 

1858 

Monday. 

C 

15 


1892 

Tuesd* 

CB 

1 

1825 

Tuesday. 

B 

11 

1859 

Tuesday. 

B 

26 


1893 

Wednes. 

A 

12 

1826 

Wednes. 

A 

22 

I860 

Thurs.* 

AG 

7 


1894 

Thursd. 

G 

23 1 

1827 

Thumb 

G 

3 

1861 

Friday. 

F 

18 


1895 

Friday. 

F 

4 

1828 

Saturd.* 

FE 

14 

1862 

Saturd. 

E 

29 


1896 

Sunday* 

ED 

15 

1829 

Sunday. 

D 

25 

1863 

Sunday. 

D 

11 


1897 

Monday. 

C 

26 

1830 

Monday. 

C 

6 

1864 

Tuesd.* 

CB 

22 


1898 

Tuesday. 

B 

7 

1831 

Tuesday. 

B 

17 

1865 

Wednes. 

A 

3 


1899 

Wednes. 

A 

18 

1832 

Thumb* 

AG 

28 

1866 

Thursd. 

G 

14 


1900 

Friday.* 

GF 

29 

1833 

Friday. 

F 

9 

1867 

Friday. 

F 

25 


1901 

Saturd. 

E 

11 


The day of the week opposite the year in the t-'manae falls on the dates in this 
table. 


February, 

February,* 


January, 

January,* 

September, 


March, 


May. 


April, 


June. 

November. 

August. 


October. 

Jul j'. 

December. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 






In leap years take January,* February.* 

Example. 1. On what day of the week will the 4th of July fall in the year 1880 ? 

See table 1880 = Monday, which answers to the 5th in the date table, conse¬ 
quently the 4th of July is on Sunday. 

Example 2. It is known to be Saturday in the middle of August, 1875. Required, 
the date of that day ? 

The year 1875 — Monday (see almanac), then August the 16th falls on Monday, i 
and Saturday on the 14tli. 

Example 3. A gentleman was born on the 8th of February, 1824. Required, 
the day of the week ? 

1824 = Monday, which fell on the 9th. The gentleman was consequently born 
on a Sunday. 


32 






































Chronology. 


498 


CHRONOLOGY. 

Our unit of time, year, is the period in which the earth makes one revolution 
around the sun, in reference to a fixed star. 

The unit do.y is the period in which the earth makes one revolution around its 
axis, in reference to the sun; but as the earth moves in an ellipse in which the 
sun is in one focus, the apparent solar day is not a constant period—that one hun¬ 
dred solar days in the winter are about half an hour longer than one hundred 
solar days in summer, for which a mean day is assumed in reference to an imagi¬ 
nary sun which falls in with the real sun about the 15th of April and 24th of De¬ 
cember, when the mean time and apparent time are alike. The mean solar day is 
divided into twenty-four hours, of which the clock indicates twelve hours twice. 

Sidereal Time. 

The Sidereal Day is the interval of time in which any fixed star passes the me¬ 
ridian. The sidereal day is only 23/n 56m. 4.09s. mean solar time, or the fixed stars 
pass the meridian, rise or set, 3m. 55.909s. earlier every day. 

A Sidereal Clock in ai* astronomical observatory marks twenty-four hours in 
the interval of time in which any fixed star passes the meridian. The Right As¬ 
cension of the heavenly bodies is the time when the body passes the meridian by 
the sidereal clock. The dial of a sidereal clock is divided into twenty-four hours, 
and the hours are numbered from one up to twenty-four. 

Years. 

The tropical year, or the periodical return of seasons, is 365.24224 days = 365 d. 
57). 48m. 49.536s. mean solar time. 

The civil year is 365 days, or nearly one-fourth of a day too short, for which one 
day is added every four years, called leap year. But this addition makes one day 
too much in every 128.866 years, which error is corrected every fourth century 
which can be divided by 4 without a remainder. 

Leap years of the Christian era are those that can be divided by 4 without .a 
remainder. 

In some countries these important corrections are not properly attended to, as 
in Russia and Greece, where the dates are now twelve days behind our Gregorian 
'reckoning. 

The wild Indians of South America reckon their time by new moons, when ail 
their festivals are celebrated. 

Oates. 

The civil date of the month commences at midnight. The astronomical date 
commences at noon. The mariner’s date (sea account) commences twelve hours 
before the civil date, and twenty-four hours before the astronomical date, or the 
mariner’s date is one day ahead of the astronomical date. 

Cycle of tlie Sun is the period of twenty-eight years, at which the days of the 
week return to the same days of the month. 

Lunar Cycle or Golden Number is the period of nineteen years, at 
which the changes of the moon fall on the same days of the month. 

To Find tlie Golden Number. 

Add 1 to the given year, divide the sum by 19, and the remainder will be the 
golden number. If 0 remains, the golden number is 19. 

The age of the Julian period on the 1st of January, 1872, is 2,404,794 days. 

Creation of tlie World. 

Creation of the world, 4000 before Christ. Julius Africanus says 5508; Samaritan 
Pentateuch, 4700; Septuagint, 5872; Josephus, 4658; Talmudists, 5344 ; and others 
give different times; but the Chinese tradition and history claim an antiquity of 
100,000 years before Christ. From geological formations, and from the working of 
rivers like-that of Niagara, and the Danube cutting through the Alps at the Iron 
Gate, it can be estimated an age of millions of years. The creation itself, or the 
accumulation of the enormous quantities of matter in the larger planets, must 
have required millions of years. No body, hovrever small, has been instantly 
created. Creation is work which consists of the three physical elements— force , 
motion and time —by which bodies grow like a tree, or by gradual accumulation of 
matter. These three physical elements constitute the trinity which governs the 
material universe. All creation, or action of whatever kind, whether mechanical, 
chemical or derived from light, heat, electricity or magnetism—all that has been 
and is to be done or undone—is accomplished by this triune function. It is omnip¬ 
otent, ubiquitous and eternal. 







Almanac. 


499 


Chronological Notes. 


Before Christ. B. C. years. 

Deluge, 2348 (Hales), . . 3154 

Tower of Babel finished, , 2247 

Chinese Monarchy, . . 2203 

Egyptian Pyramids, . . . 2090 

Moses born, .... 1567 

Troy destroyed, .... 1180 

The compass discovered, . 1111 

Foundation of Rome, . . 753 

Maps and Geometry introduced, 605 
Money coined at Rome, . . 576 

Hannibal crossed the Alps, . 219 

Time first measured by water, 155 

Carthage destroyed, . . 146 

Caesar invaded Britain, . . 51 
Caesarean era, ... 48 


After Christ. A. D. 

Beginning of Christian Era, 
Christ crucified, 

Destruction of Jerusalem. . 
Arabic numbers introduced, 
Mohammedan Era, 

New Style in England, . 
America discovered, . 

Pizarro conquered Inca, Peru, 
Lutheran religion, . 

New South Wales discovered, 
Australia discovered,. 
American great Republic, . 
Slaves free in West Indies, 
Slaves free in Russia, 

Slaves free in America, 


YEARS. 

0 

37 

69 

991 

622 

1752 

1492 

1530 

1527 

1606 

1622 

1775 

1834 

1861 

1866 


ASTRONOMICAL ALMANAC. 


From the English Nautical Almanac. 

The following tables show the sun’s right ascension and declination ; also, the 
equation of time at Greenwich, apparent noon, for the year 1873: 

In leap years ..at 6 h. A. M. 

First year after leap year.at noon. 

Second year after leap year.at 6 h. P. M. 

In the year before leap year, at midnight following the date. 

By the aid of the tables of correction the data can be found for any time and 
for any meridian. 


Example 1. Required, the sun’s R. A. on the 10th of April, 1874, at Greenwich, 
apparent noon? 

1874 is the second year after leap year, when the tabular data is for 6 o’clock in 
the evening. The daily variation of the sun’s R. A. is 3 m. 40 s. for the 10th of 
April, which for 6 hours will be 55 s. 

The sun’s R. A. on the 10th of April, . 1 h. 16 m. 24 s. 

Correction for 6 hours, subtract . . ._ 55 s. 

The required R. A. will be . . . 1 h. 15 m. 29 s. 

Example 2. Required, the sun’s declination on the 20th of September, in the 
leap year 1876, at 3 o’clock P. M., in longitude 75°, or 5 hours west of Greenwich ? 

The tabular data for leap years is at 6 o’clock A. M. 3 P. M. and 5 hours differ¬ 
ence in longitude make 14 hours of correction. 

The daily variation in the sun’s declination is 23'. 


20' = 11' 40") 
3 = 1 45 | 


See table of correction, page 502. 


Correction, 13' 25" for 14 hours. 


Sun’s declination 20th September, ... 0° 58' 

Subtract correction, . . ... 13' 25" 

The required declination,.0° 44' 35" 

In leap years take the tabular data one day earlier in January and February. 







Astronomical Almanac. 


500 



January 


February. 

March 

• 


Date 

fit. As. 

Declin. 

Eq. tm. 

Rt. As. 

Declin. 

Eq.tin. 

Rt. As. 

Deciiu. 

Eq. tin. 

Date 


h. m. s. 

O ' 

m. s. 

h. m. s. 

O * 

111. s. 

h. m. s. 

O ' 

m. s. 


1 

18 48 49 

S 22 59 

+ 3 59 

21 0 59 

S 16 59 

+13 54 

22 49 58 

S 716 

+12 30 

1 

2 

18 53 14 

S 22 54 

+ 4 27 

21 5 3 

S 16 42 

+14 2 

122 53 42 

S 7 3 

+12 18 

2 

3 

18 57 38 

S 22 48 

+ 4 55 

21 9 6 

S 16 24 

+14 8 

|22 57 26 

S 6 40 

+12 5 

3 

4 

19 2 2 

S 22 41 

+ 5 23 

2113 8 

S 16 6 

+1414 

23 1 9 

S C17 

+11 52 

4 

5 

19 6 26 

S 22 35 

+ 5 49 

2117 9 

S 15 48 

+ 1418 

23 4 62 

8 5 54 

+11 38 

5 

6 

19 10 49 

S 22 27 

+ 616 

21 2110 

S 15 30 

+14 22 

23 8 35 

S 5 31 

+11 24 

6 

7 

19 15 11 

S 22 20 

+ 6 42 

12125 9 

S 1511 

+14 25 

123 12 17 

S 5 8 

+11 10 

7 

8 

19 19 33 

S 22 12 

+ 7 7j 

,2129 8 

S 14 52 

+14 28 1 

|23 15 58 

S 4 44 

+10 55 

8 

9 

19 23 55 

S 22 3 

+ 7 32, 

,2133 6 

S 14 33 

+ 14 29 

2319 39 

S 4 21 

+10 39 

9 

10 

19 2816 

S 2154 

+ 7 56. 

'2137 3 

S 1413 

+14 30 

23 23 20 

S 3 57 

+10 24 

10 i 

11 

19 32 36 

S 2145 

+ 8 20' 

21 41 0 

S 13 53 

+14 30 

23 27 1 

S 3 84 

+10 7 

11 1 

12 

19 36 56 

S 2135 

+ 8 43 

21 44 56 

S 13 33 

+ 14 29 

23 30 41 

S 310 

+ 9 51 

12 1 

13 

19 4115 

S 2125 

+ 9 6 

,2148 51 

S 13 13 

+14 27 

23 34 21 

S 2 47 

+ 9 34 

13 

14 

19 45 33 

S 21 14 

+ 9 27 

[21 52 45 

S 12 53 

+14 25 

23 38 0 

S 2 23 

+ 9 17 

14 

15 

19 49 51 

8 21 3 

+ 9 48 

21 56 38 

S 12 32 

+ 14 22 

23 41 39 

S 159 

+ 90 

15 

16 

19 54 8 

S 20 52 

+10 9 

j22 0 31 

S 1212 

+14 18 

23 45 19 

S 136 

+ 8 43 

16 

17 

19 58 24 

8 20 )0 

+10 29 

22 4 23 

8 11 51 

+14 13 

23 48 57 

S 112 

+ 8 25 

17 

18 

20 2 40 

S 20 28 

+10 48 

22 814 

S 1130 

+ 14 8 

23 52 36 

S 0 48 

+ 88 

18 

19 

20 6 55 

S 2015 

+11 6 

'2212 5 

S 11 8 

+14 2 

23 56 15 

S 0 24 

+ 7 50 

19 

20 

20 11 9 

8 20 2 

+11 24 

22 15 55 

8 10 47 

+13 56 

23 59 53 

SOI 

+ 7 32 

20 

21 

20 15 23 

S 19 49 

+11 40 

22 19 44 S 10 25 

+ 13 49 

0 3 32 

X 0 23 

+ 713 

21 

22 

20 19 35 

8 19 35 

+11 57 

22 23 33 S 10 3 

+ 13 41 

0 710 

X 0 47 

+ 6 55 

22 

23 

20 23 47 

S 19 21 

+1212 

22 27 31 

S 9 41 

+ 13 32 

0 10 48 

X 1 10 

+ 6 37 

23 

24 

20 27 59 

S 19 7 

+12 27 

22 31 9 

S 919 

+13 23 

014 26 

X 134 

+ 618 

24 

25 

20 32 9 

S 18 52 

+12 40 

22 3156 

S S 57 

+13 14 

018 4 

X 157 

+ 60 

25 

26 

20 36 19 

S 18 37 

+12 53 

22 38 42 

S 8 34 

+13 4 

0 21 42 

X 2 21 

+ 5 42 

26 

27 

20 40 27 

S 18 21 

+13 6 

22 42 28 

S 812 

+12 53 

0 25 2 i 

X 2 14 

+ 5 23 

27 

28 

20 44 35 

8 18 6 

+ 13 17' 

22 46 13 

S 7 49 

+12 42 

0 28 59 

X 3 8 

-)- 5 5 

28 

29 

20 48 43 

8 17 49 

+13 23 




0 32 37 

X 3 31 

+ 4 47 

29 

30 

20 52 49 

S 17 33 

+13 37 




0 86 15 

X 3 55 

+ 4 28 

30 

31 

20 56 54 

8 17 16 

+ 13 46 




0 39 53 

X 4 18 

+ 410 

31 

Date 

April. | 

May. | 

June. 

Date 

1 

0 43 32 

N 4 41 

+ 352' 

2 34 51 

N 15 11 

— 34 

4 37 41 

X 22 6 

— 2 27 

1 

2 

0 4710 

X 5 4 

+ 3 34' 

2 38 41 

N 15 29 

— 3 11 

4 41 47 

X 22 14 

— 2 18 

• > 

3 

0 50 49 

N 527 

+ 3 16 

2 42 31 

X 15 57 

— 318 

4455! 

X 2 1 22 

— 28 

3 

4 

0 54 27 

N 5 50 

+ 2 58 

2 46 21 

NIG 4 

— 321 

4 50 0 

X 22 29 

— 158 

4 

5 

0 58 6 

N 6 13 

+ 2 41 

2 5012 

N 16 21 

— 3 29 

4 54 7 X22 38 

— 148 

5 

6 

1 145 

N 6 35 

+ 2 23 

2 54 4 

X 16 38 

— 3 34 

4 58 14 

X 22 42 

— 1 37 

6 

7 

1 5 25 

X 6 58 

+ 26 

2 57 56 

N 16 55 

— 3 39 

5 2 22 

X 22 48 

— 1 27 

7 

8 

19 4 

N 7 20 

+ 149 

3 1 49 

N 17 11 

— 3 42 

5 6 29 

X 22 53 

— 1 15 

8 

9 

112 44 

N 7 43 

+ 132 

3 5 42 

N 17 27 

— 3 46 

5 10 37 

X 22 58 

— 14 

9 

10 

116 24 

X 8 5 

+ 1 15 

3 9 36 

X 17 43 

— 3 48 

5 14 46 1 X 23 3 

— 0 52 

10 

11 

120 4 

X 8 27 

+ 0 59 

3 13 30 

N 17 58 

— 3 51 

5 18 54 

X 23 7 

— 0 40 

11 

12 

1 23 44 

N 8 49 

+ 0 43 

3 17 25 

N 18 13 

— 3 52 

52 5 3 

X 23 11 

— 3 28 

12 

13 

1 27 25 

X 911 

+ 0 27 

3 21 21 

X 18 28 

— 3 53 

5 27 12 

X 23 15 

— 016 

13 

14 

131 6 

X 9 32 

+ 012 

3 25 17 

X 18 43 

— 3 54 

5 31 21 

X 2318 

— 04 

14 

15 

134 48 

N 954 

— 0 3] 

3 29 13 

N 18 57 

— 3 53 

5 35 301X 23 20 

+ 09 

15 

16 

138 30 

N 10 15 

— 018' 

3 3311 

X 19 11 

— 3 53 

5 39 39 

X 23 22 

+ 0 22 

m 

17 

14212 

X 10 36 

— 0 32j 

3 37 9 

N 19 24 

— 3 51 

5 43 49 

X 23 21 

+ 0 35 

17 1 

18 

1 45 55 

N 10 57 

— 0 46 

3 41 7 

N 19 38 

— 3 50 

5 47 58' 

X 23 26 

+ 0 48 

is 

19 

14)38 

N 11 18 

— 0 59 

3 45 6 

N 19 51 

— 3 47 

5 52 8 

X 23 27 

+ 1 1 

19 

20 

1 53 21 

N T 11 38 

— 1 12 

3 49 6 

X 20 3 

— 344 

5 £6 17 i X 23 27 

+ 114 

20 

21 

157 5 

X 11 59 

- 125' 

3 53 6 

X 20 15 

— 3 40 

6 0 27 

X 23 27 

+ 127 

21 

22 

2 0 49 

X 1213 

— 1 37! 

3 57 6 

X 20 27 

— 3 36 

6 4 37 

X 23 27 

+ 1 40 

22 

23 

2 4 34 

N 12 39 

— 1 49 

4 18 

N 20 39 

— 3 32 

6 8 46 

X 23 26 

\- 1 c3 

23 

24 

2 8 20 

N 12 59 

— 20 

4 510 

N 20 50 

— 3 26 

612 5ii 

X 23 25 

+ 26 

24 

25 

2 12 5 

N 13 18 

— 211 

4 9 12 

X 21 1 

— 3 21 

6 17 5 

X 23 24 

+ 2 19 

25 

26 

2 15 52 

X 13 38 

— 2 21] 

4 13 15 

X 21 11 

— 315 

6 21 15 

X 23 22 

+ 2 31 

26 

27 

2 19 39 

N 13 57 

— 2 34 

4 17 18 

X 21 22 

— 38 

6 25 24 

X 23 20 

+ 2 44 

27 

28 

2 23 26 

N 1416 

— 2 40; 

i 4 21 22 

X 21 31 

— 3 0 

6 29 33 

X 2317 

+ 2 56 

28 

29 

2 27 14 

N 14 34 

— 2 48 ] 

4 25 26 

X 21 41 

— 2 53 

6 2.3 42 

X 2314 

+ 38 

2 1 

30 

2 31 2 

X 14 53 

— 266' 

1 4 29 31 

X 21 50 

— 2 45 

6 37 50 

X -.3 10 

+ 3 20, 

Mil 

31 



4 

1 4 33 361 N 21 58 

— 2 36 



1 

31 












































































Astronomical Almanac. 


■ 

July. 

August 


September. 

Date 

Rt. As. 

Declin. 

Eq. tm.] 

Rt. As. 

Declin. 

Eq. tmj 

Rt. As. 

Declin. 

Eq.tin 


h. m. s. 

O ' 

Ill. s. 

h. m. s. 

O ' 

m. s. i 

h. m. <s. 

O ' 

m. s. 

1 

6 4158 

N 23 6 

+ 3 32 

8 46 42 

N 17 57 

+ 62 

10 42 40 

N 810 

— 012 

2 

6 46 6 

N 23 2 

+ 3 44. 

8 50 35 

N 17 42 

+ 5 58 i 

10 46 18 

N 7 49 

— 081 

3 

6 50 14 

N 22 57 

+ 3 55 

8 54 27 

N 17 26 

+ 5 54 

10 49 65 

N 7 27 

— 0 50 

4 

6 54 22 

N 22 52 

+ 45 

8 58 18 

N 1711 

+ 5 49 ; 

ilO 53 32 

N 7 4 

— 110 

5 

6 58 29 

N 22 47 

+ 416 

9 2 9 

N 16 54 

+ 5 43 

10 57 8 

N 6 42 

— 130 

6 

7 2 35 

N 22 41 

+ 4 26 

9 5 5 .) 

N 16 38 

+ 5 36 1 

11 0 45 

N 6 20 

— 150 

7 

7 6 42 

N 22 34 

+ 4 36, 

9 9 49 

N 16 21 

+ 5 29 

11 421 

N 5 57 

— 210 

8 

710 48 

N 22 28 

+ 4 45 

913 38 

N 16 4 

+ 5 22 

11 7 57 

N 5 35 

— 2 3) 

9 

7 14 53 

N 22 20 

+ 4 54 

917 26 

N 15 47 

+ 514 

1111 33 

N 512 

— 2 511 

10 

718 58 

N 2213 

+ 53 

9 2114 

N 15 29 

-f- 5 5 

1115 9 

N 449 

— 312 

11 

7 23 3 

N 22 5 

+ 511 

9 25 1 

N 15 12 

-f~ 4 55 

11 18 45 

N 4 27 

— 3 33 

12 

7 27 7 

N 2157 

+ 5 18 1 

9 28 47 

N 14 54 

+ 4 46 

11 22 20 

N 4 4 

— 3 54 

13 

7 3111 

N 21 48 

+ 5 26 

9 3234 

N 14 35 

+ 4 35 

11 25 56 

N 3 41 

— 415 

14 

7 35 14 

N 21 39 

+ 5 32j 

9 36 19 

N 1417 

+ 424 

1129 31 

N 318 

-4 36 

15 

7 39 17 

N 21 30 

+ 5 39 1 

9 40 4 

N 13 58 

+ 413 

11 33 6 

N 2 54 

— 457 

16 

7 4319 

N 21 20 

+ 5 44] 

9 43 49 

N 13 39 

+ 41 

11 36 42 

N 2 31 

— 5 18 

17 

7 47 21 

N 21 10 

+ 5 50. 

9 47 33 

N 13 20 

+ 3 48 

11 40 17 

N 2 8 

— 6 S9 

IS 

7 51 23 

N 20 59 

+ 5 54 

9 5116 

N 13 1 

+ 3 35 

11 43 53 

N 145 

— 60 

19 

7 55 21 

N 20 49 

+ 5 59] 

9 54 59 

N 12 41 

+ 3 22 

111 47 28 

N 122 

— 6 21 

20 

7 59 24 

N 20 37 

+ 6 2] 

9 58 42 

N 12 21 

+ 38 

1151 4 

N 0 58 

— 6 42 

21 

8 3 24 

N 20 26 

+ 6 5 ] 

10 224 

N 12 1 

+ 2 54 

1154 39 

N 0 35 

- 7 3 

22 

8 7 23 

N 2014 

+ 6 8i 

10 6 6 

N 11 41 

+ 2 39 

1158 15 

N 011 

— 7 24 

23 

811 22 

N 20 2 

+ 6 10 

10 9 47 

N 11 21 

+ 2 24 

12 1 51 

S 012 

— 7 45 

24 

815 20 

N 19 49 

+ 612 

10 13 2S 

Nil 0 

+ 28 

12 5 26 

S 0 35 

— 85 

25 

8 1917 

N 19 36 

+ 613 

1017 8 

N 10 40 

+ 1 52 

12 9 3 

S 0 59 

— 8 26 

26 

8 23 14 

N 19 23 

+ 613] 

10 20 48 

N 10 19 

+ 135 

12 12 39 

S 122 

— 8 46 

27 

8 27 10 

N 19 10 

+ 6 131 

10 24 28 

N 9 58 

+ 118 

121615 

S 146 

— 96 

28 

8 31 6 

N 18 56 

+ 612 

10 28 7 

N 9 37 

+ 1 1 

12 19 62 

S 2 9 

— 9 26 

29 

8 35 1 

N 18 42 

+ 611 

10 3146 

N 915 

+ 0 43 

12 23 29 

8 2 32 

— 9 46 

30 

8 38 55 

N 18 27 

+ 69 

10 35 24 

N 8 54 

+ 0 25 

12 27 6 

S 2 56 

—10 5 

31 

8 42 49 

N 18 13 

+ 6 6] 

10 39 2 

N 8 32 

+ 07 




Date 

October. 

November. 

| December. 

1 

12 30 43 

S 3 19 

—10 24s 

14 27 2 

S 14 33 

—1617 ! 

16 30 57 

8 21 53 

—10 40 

2 

12 34 20 

S 3 42 

—10 431 

14 30 58 

8 14 52 

—1618 

116 35 17 

S 22 2 

—10 17 

3 

12 37 58 

S 4 6 

—11 2 

1434 54 

S 1511 

—16 18 

16 39 37 

S 2210 

— 9 53 

4 

12 41 37 

S 4 29 

—1120 

44 38 52 

S 15 30 

—16 4 7 

16 43 58 

S 2218 

— 9 29 

5 

12 45 15 

S 4 52 

—11 881 

14 42 50 

8 15 38 

—1615 

16 48 20 

S 22 26 

— 94 

6 

12 48 54 

S 515 

—1156 

14 46 49 

S 16 6 

—1613 

16 52 42 

S 22 33 

— 8 38 

7 

12 52 33 

S 5 38 

—1213 

44 50 4-9 

S 16 24 

—1610 

1657 4 

S 22 40 

— 813 

8 

12 56 13 

S 6 1 

—12 29 

,14 54 50 

S 16 41 

—16 6 

,17 127 

8 22 46 

— 7 46 

9 

12 59 53 

S 6 24 

—12 46 

14 58 51 

S 16 59 

—16 1 

117 5 51 

S 22 52 

— 7 19 

10 

13 3 34 

S 6 47 

—13 2 

15 2 54 

S 17 16 

—15 55 

17 1015 

S 22 58 

— 6 52 

11 

13 715 

S 7 9 

—13 171 

15 6 57 

S 17 32 

—15 48 

17 1439 

S 23 3 

— 624 

12 

13 10 57 

S 7 32 

—13 32 ! 

1511 1 

8 17 48 

—15 40 

1719 4 

S 23 7 

— 5 56 

13 

13 14 39 

S 7 54 

—13 46 

1515 6 

S 18 4 

—15 32 

117 23 29 

S 2312 

— 5 28 

14 

1318 22 

S 817 

—14 0 

151912 

S 18 20 

—1522 

17 27 54 

S 2315 

— 4 59j 

15 

13 22 5 

S 8 39 

—1413 

15 2319 

S 18 36 

—1512 

117 32 20 

S 23 18 

— 4 30 

16 

13 25 49 

S 9 1 

—14 26 

15 27 27 

S 18 51 

—15 1 

117 36 46 

8 23 21 

— 41 

17 

13 29 34 

S 9 23 

-—14 38 

15 31 35 

S 19 6 

—14 49 

17 4112 

S 23 23 

— 3 31 

18 

13 33 19 

S 9 45 

—14 49 

15 35 45 

S 19 20 

—14 36 

17 45 38 

S 23 25 

— 31 

19 

13 37 4 

S 10 7 

—15 0 

15 39 55 

S 19 34 

—14 23 

17 50 5 

S 23 26 

— 2 32 

20 

13 40 51 

S 10 28 

—1510 

|15 44 6 

S 19 47 

—14 8 

117 54 32 

S 23 27 

_ 2 •/ 

21 

13 44 38 

S 10 50 

—1519 

.15 4818 

8 20 1 

—13 53 

17 58 58 

S 23 27 

— 131 

22 

13 48 26 

S 11 11 

—15 28 

15 52 30 

S 20 14 

—13 37 

18 3 25 

8 23 27 

— 11 

23 

13 52 14 

S 11 32 

—15 36 

Jl 5 56 44 

S 20 26 

—13 20 

18 7 52 

S '.3 27 

— 0 31 

24 

1356 3 

S 11 53 

—15 44 

|16 0 58 

S 20 38 

—13 3 

18 12 is 

S 23 26 

— 0 1 

25 

13 59 53 

S 1214 

—15 51 

116 5 13 

S 20 50 

— 12 44 

18 16 45 

8 28 24 

+ 0 29 

26 

14 3 43 

S 12 34 

—15 57 

.16 9 29 

S 21 2 

—12 25 

18 2111 

S 23 22 

+ 0 5S 

27 

14 7 35 

S 12 55 

—16 2 

jl613 45 

S 2113 

—12 5 

18 25 38 

8 23 20 

+ 128 

28 

1411 27 

S 13 15 

—16 7 

16 18 2 

S 21 23 

—11 45 

18 30 4 

S 2317 

-\- 1 58 

29 

1415 19 

S 13 35 

—1610 

,16 22 20 

S 21 34 

—11 24 

118 3130 

S 2313 

+ 227 

30 

1419 13 

S 13 55 

—1614" 

10 26 38 

S 21 43 

— 11 2 

18 38 55 

S 23 9 

+ 2 56 

31 

14 23 7|S 14 14 

—16161 

1 



118 43 21 

S 23 5 

+ 3 24' 





























































502 


Astronomy. 


Corrections in Minutes and Seconds of Eight Ascension, 
Declination and Equation, of Time for Hours up to 18. 


D0 

Sm 







Variations 

IN 

Seconds for 24 

Hours. 






tn 

CD 

0> 






















bh 

1—i 

5 

6 


7 


8 

9 

10 

15 

20 

35 

30 

35 

40 

45 

50 

55 

ft 


ft 

tt 

ff 


tt 

tf 

tf 

tt 

tt 

ft 

tt 

tf 

ff 

tf 


ft 

tf 



1 

0 

0 


0 


0 

0 

0 

1 

1 

1 

1 


1 

2 

2 


2 

2 


15 

2 

0 

1 


1 


1 

1 

1 

1 

2 

2 

3 


3 

3 

4 


4 

5 


30 

3 

1 

1 


1 


1 

1 

1 

2 

3 

3 

4 

4 

5 

6 


6 

7 


45 

4 

1 

1 


1 


1 

2 

2 

3 

3 

4 

5 


6 

7 

7 


8 

9 


60 

5 

1 

1 


1 


2 

2 

2 

3 

4 

5 

6 


7 

8 

9 

10 

11 


75 

6 

1 

2 


2 


2 

2 

3 

4 

5 

6 

8 


9 

10 

11 

12 

14 


90 

7 

1 

2 


2 


2 

3 

3 

4 

6 

7 

9 

10 

12 

13 

16 

16 


105 

8 

2 

2 


2 


3 

3 

3 

5 

7 

8 

10 

12 

13 

15 

17 

18 


120 

9 

2 

2 


3 


3 

3 

4 

6 

8 

9 

11 

13 

15 

17 

19 

21 


135 

10 

2 

3 


3 


3 

4 

4 

6 

8 

10 

13 

15 

17 

19 

21 

23 


150 

11 

2 

3 


3 


4 

4 

5 

7 

9 

11 

14 

16 

18 

21 

23 

25 


165 

12 

3 

3 


4 


4 

5 

5 

8 

10 

13 

15 

17 

20 

22 

25 

27 


180 

13 

3 

3 


4 


4 

5 

5 

8 

11 

14 

16 

19 

23 

24 

27 

30 


195 

14 

3 

4 


4 


5 

5 

6 

9 

12 

15 

18 

20 

23 

26 

29 

32 


210 

15 

3 

4 


4 


5 

6 

6 

9 

13 

16 

19 

22 

25 

28 

31 

34 


225 

16 

3 

4 


5 


5 

6 

7 

10 

13 

17 

20 

23 

27 

30 

33 

37 


240 

17 

4 

4 


5 


6 

6 

7 

11 

14 

18 

21 

25 

28 

32 

35 

39 


255 

18 

4 

5 


5 


6 

7 

8 

11 

15 

19 

23 

26 

30 

34 

37 

41 


270 

ro 







Variation in Minutes for 24 Hours. 






CO 

03 

03 

53 





















So 

W 

1 



3 


3 

4 

5 

G 


7 

8 



9 

10 


20 


ft 





ft 

t 

// 

' •/ 

/ tt 


tt 

t tt 

/ // 

/ 

tt 

t ft 


/ 

tt 



1 


2 


5 


7 

10 

12 

15 

17 

20 


22 

25 


50 


15 

2 


5 


10 


15 

20 

25 


£0 

35 

40 


45 

50 

1 

40 


30 

3 


7 


15 


22 

30 

37 

45 

52 

1 


1 

7 

1 15 

2 

30 


45 

4 

10 


20 


30 

40 

50 

1 


1 10 

1 20 

1 

£0 

1 40 

3 

20 


60 

5 

12 


25 


S7 

50 

1 2 

1 

J5 

1 27 

1 40 

1 

62 

2 5 


4 

10 


75 

6 

15 


30 


45 

1 

1 15 

1 

30 

1 45 

2 


2 

15 

2 30 

5 



90 

7 

17 


35 


52 

1 10 

1 27 

1 

45 

2 2 

2 2 

0 

2 

37 

2 55 

5 

50 

105 

8 

20 


40 

1 


1 20 

1 40 

2 


2 20 

2 40 

3 


3 20 

6 

40 

120 

9 

22 


46 

1 

7 

1 30 

1 52 

2 

15 

2 37 

3 


3 

22 

3 45 

7 

30 

135 

10 

26 


50 

1 

15 

1 40 

' 2 5 

2 

30 

2 55 

3 20 

3 

45 

4 10 

8 

20 

150 

11 

27 


55 

1 

22 

1 50 

2 17 

2 

46 

3 12 

3 40 

4 

7 

4 35 

9 

10 

166 

12 

30 

1 


1 

30 : 2 

2 30 

3 


3 30 

4 


4 

30 

5 


10 


180 

13 

O 

o 

2 

1 

5 

1 

37 

2 10 

2 42 

3 

15 

3 47 

4 20 

4 

52 

5 25 

lu 

50 

195 

14 

35 

1 

10 

1 

45 

2 20 

2 55 

3 

30 

4 5 

4 40 

5 

15 

5 50 

11 

40 

210 

15 

37 

l 

15 

1 

52 

2 30 

3 7 

3 

45 

4 22 

5 


5 

37 

6 15 

12 

30 

225 

16 

40 

1 

20 

2 


2 40 

3 20 

4 


4 40 

5 20 

6 


6 40 

13 

20 

240 

17 

42 

1 

25 

2 

7 

2 £0 

3 32 

4 

15 

4 57 

5 40 

6 

22 

7 5 

14 

10 

255 

18 

45 

1 

3© 

2 

15 

3 

3 45 

4 

30 

5 15 

6 


6 

45 

7 30 

15 


270 


Explanation of tlie Sidereal and Solar Time Table. 

The Sidereal Time — Mean Solar + Correction. 

Mean Solar Time = Sidereal — Correction. 

To Find tlie True Sidereal Time. 

The sun’s Right Ascension -f or— the Equation of time is the Sidereal time at 
Greenwich, mean noon. 

The sign + or — must be used as noted in the Astronomical or Nautical Almanac 
for the given day. For any other meridian or longitude from Greenwich, cor¬ 
rect the sun’s R. A. and the equation of time, and perform the same operation. 






































































Refraction of the Heavenly Bodies in Altitude. 


503 


Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

D.M. 

M. S. 

D.M. 

M. S. 

D. M. 

M. S. 

D. M. 

M. S. 

D. 

M. S. 

D. 

M. S. 

0. 0 

33. 0 

2.30 

16.23 

6.30 

7.52 

12.20 

4.16 

30 

1.38 

60 

0.33 

0. 5 

32.11 

2.35 

16. 4 

6.40 

7.41 

12.40 

4. 9 

31 

1.35 

61 

0 32 

0.10 

31.22 

2.40 

15 45 

6.50 

7.31 

13. 0 

4. 3 

32 

1.31 

62 

0.30 

0-15 

30.36 

2.45 

15.27 

7. 0 

7.21 

13.20 

3.57 

33 

1.28 

63 

0.29 

0.20 

29.50 

2.50 

15. 9 

7.10 

7.12 

13.40 

3.51 

34 

1.24 

64 

0.28 

0.25 

29. 6 

2.55 

14.52 

7.20 

7. 3 

14. 0 

3.46 

35 

1.21 

65 

0.27 

0.30 

28.23 

3. 0 

14.35 

7.30 

6.54 

14.20 

3.40 

36 

1.18 

66 

0.25 

0.35 

27.41 

3. 5 

14.19 

7.40 

6.46 

14.40 

3.35 

87 

1.16 

67 

0.24 

0.40 

27. 0 

3.10 

14.03 

7.50 

6.38 

15. 0 

3.30 

38 

1.13 

68 

0.23 

0.45 

2G.20 

3.15 

13.48 

8. 0 

6.30 

15.30 

3.23 

39 

1.10 

69 

0.22 

0.50 

25.42 

3.20 

13.33 

8.10 

6.22 

16. 0 

3.17 

40 

1. 8 

70 

0.21 

0.55 

25. 5 

3.25 

13.19 

8.20 

6.15 

16.30 

3.11 

41 

1. 5 

71 

0.20 

1. 0 

24.29 

3.30 

13.05 

8.30 

6. 8 

17. 0 

3. 5 

42 

1. 3 

72 

0.19 

1. 5 

23.54 

3.40 

12.39 

8.40 

6. 1 

17.30 

2.59 

43 

1. 1 

73 

0.17 

1.10 

23.20 

3.50 

12.14 

8.50 

5.55 

18. 0 

2.54 

44 

0.59 

74 

0.16 

1.15 

22.47 

4. 0 

11.50 

9. 0 

5.49 

18.30 

2.49 

45 

0.57 

75 

0.15 

1.20 

22.15 

4.10 

11.28 

9.10 

5.43 

19. 0 

2.44 

46 

0.55 

76 

0.14 

1.25 

21.44 

4.20 

11.07 

9.20 

5.37 

19.30 

2.40 

47 

0.53 

77 

0.13 

1.30 

21.15 

4.30 

10.47 

9.30 

5.31 

20. 0 

2.36 

48 

0.51 

78 

0.12 

1.35 

20.46 

4.40 

10.28 

9.40 

5.26 

20.30 

2.32 

49 

0.50 

79 

0.11 

1.40 

20.18 

4.50 

10.10 

9.50 

5.20 

21. 0 

2.28 

50 

0.48 

80 

0.10 

1.45 

19.51 

5. 0 

9.53 

10. 0 

5.15 

21.30 

2.24 

51 

0.46 

81 

0. 9 

1.50 

19.25 

5.10 

9.37 

10.15 

5. 8 

22. 0 

2.20 

52 

0.45 

82 

0. 8 

1.55 

18.59 

5.20 

9.21 

10.30 

5. 0 

23. 0 

2.14 

53 

0.43 

83 

0. 7 

2. 0 

18.35 

5.30 

9. 7 

10.45 

4.54 

24. 0 

2. 7 

54 

0.41 

84 

0. 6 

2. 5 

18.11 

5.40 

8.53 

11. 0 

4.47 

25. 0 

2. 2 

55 

0.40 

85 

0. 5 

2.10 

17.48 

5.50 

8 39 

11.15 

4 41 

26. 0 

1.56 

56 

0.38 

86 

0. 4 

2.15 

17.26 

6. 0 

8.27 

11.30 

4.35 

27. 0 

1.51 

57 

0.37 

87 

0. 3 

2.20 

17. 4 

6.10 

8.15 

11.45 

4.29 

28. 0 

1.47 

58 

0.36 

88 

0. 2 

2.25 

16.44 

6.20 

8. 3 

12. 0 

4.23 

29. 0 

1.43 

59 

0.34 

89 

0. 1 


Conversion of Sidereal «fc Mean Solar Times. 


Hour. 

Corr. 

Mia. 

Corr. 

Min. 

Corr. 

Sec. 

Corr. 

Sec. 

Corr 

H. 

M. S. 

M. 

S. 

M. 

S. 

S. 

S. 

s. 

S. 

1 

0 9.8 

1 

0.2 

31 

5.1 

1 

o-o 

31 

0.1 

2 

0 19.7 

2 

0.3 

32 

5.2 

2 

0.0 

32 

0.1 

3 

0 29.5 

3 

0.5 

33 

5.4 

3 

0.0 

33 

0.1 

4 

0 39.3 

4 

0.7 

34 

5.6 

4 

0.0 

34 

0.1 

5 

0 49.1 

5 

0.8 

35 

5.7 

5 

0.0 

35 

0.1 

6 

0 59.0 

6 

1.0 

36 

5.9 

6 

0.0 

36 

0.1 

7 

1 8.9 

7 

1.1 

37 

6.1 

7 

0.0 

37 

0.1 

8 

1 18.7 

8 

1.3 

38 

6.2 

8 

0.0 

38 

O.i 

9 

1 28.6 

9 

1.5 

39 

6.4 

9 

0.0 

39 

0.1 

10 

1 38.4 

10 

1.6 

40 

6.6 

10 

0.0 

40 

0.1 

11 

1 48.2 

11 

1.8 

41 

6.7 

11 

0.0 

41 

0.1 

12 

1 58.1 

12 

2.0 

42 

6.9 

12 

0.0 

42 

0.1 

13 

2 8.0 

13 

2.1 

43 

7.0 

13 

0.0 

43 

0.1 

14 

2 17.8 

14 

2.3 

44 

7.2 

14 

0.0 

44 

0.1 

15 

2 27.6 

15 

2.5 

45 

7.4 

15 

0.0 

45 

0.1 

16 

2 37.5 

16 

2.6 

46 

7.5 

16 

0.0 

46 

0.1 

17 

2 47.3 

17 

2.8 

47 

7.7 

17 

0.0 

47 

0.1 

18 

2 57.1 

18 

2.9 

48 

7.9 

18 

0.0 

48 

0.1 

19 

3 7.0 

19 

3.1 

49 

8.0 

19 

0.1 

49 

0.1 

20 

3 16.9 

20 

3.3 

50 

8.2 

20 

0.1 

50 

0.1 

2L 

3 26.7 

21 

3.4 

51 

8.4 

21 

0.1 

51 

0.1 

22 

3 36.5 

22 

3.6 

52 

8.5 

22 

0.1 

52 

0.1 

23 

3 46.4 

23 

3.8 

53 

8.7 

23 

0.1 

53 

0.1 

24 

3 56.3 

24 

3.9 

54 

8.8 

24 

0.1 

54 

0.1 



25 

4.1 

55 

9.0 

25 

0.1 

55 

0.2 



26 

4.3 

56 

9.2 

26 

0.1 

56 

0.2 



27 

4.4 

57 

9.3 

27 

0.1 

57 

0.2 



28 

4.6 

58 

9.5 

28 

0.1 

58 

0.2 



29 

4.8 

59 

9.7 

29 

0.1 

59 

0.2 



30 

4.9 

60 

9.8 

30 

01 

60 

0.2 


Tlie Sim’s Parallax 
in Altitude. 


Altitude. 


Parallax. 


D. 

0 

10 

20 

30 

40 

50 

55 

60 

65 

70 

75 

80 

85 

90 


5 

9 

9 

8 

8 

7 

6 
5 
4 
4 
3 
2 
2 
1 
0 


Explanation. 

The sun’s parallax 
must be added to the 
observed altitude. 
































504 


Astronomy. 


LATITUDE AND APPARENT TIME 


By Altitude of the Heavenly Bodies. 

Notation of Letters. 


A = meridian altitude above horizon. 

D = declination, to be found in the Astronomical Almanac. 

I — latitude of the place of observation. 

L = angle of apparent time from noon. 
a - any altitude of the heavenly body, before or after noon. 


When the, latitude and declination are of 

Equal Name. 

Latitude, l = 90 -f- D — A. 
Altitude, A — 90 + D — l. 

Declination, D = A + l — 90. 

Apparent Time, 

Cos.L = sin.a sec.J sec.Z> — tan.? tan. D. 


When the latitude and declination are of 

Different Names. 

Latitude, l — 90 — A — D. 
Altitude, A = 90 — l — D. 
Declination, D = 90 — A — l. 

Apparent Time, 

Gos.L == sin.asecisec.D + tani tan.Z>. 


At sea the altitude is observed from the visible horizon of the ocean, from which 
must be subtracted the dip of horizon. (See table, page 131.) 

On land the horizon must be determined by a spirit-level, or more correctly by 
an artificial horizon of quicksilver, oil, syrup or some similar liquid. 

The refraction of light through the atmosphere (see table, page 503) must also be 
subtracted from the observed altitude. 


When the sun or moon is observed, the parallax must be added to the observed 
altitude. 


Tati tnde. 


Example 1. On the 7th of April, 1872, the sun’s lower limb was observed to be 
51° 42' 50" above the horizon at noon, in longitude about 45° west of Greenwich; 
the observation was made from the deck of a vessel 20 feet above the sea. Required, 
the sun's true altitude and latitude of observation? 


The declination and latitude are both north or of equal name. 


Dip of horizon for 20 feet, 4' 24" 

Refraction, 51°, .... 46 

O’s semi-diameter, . . . 16 00 

' 21 10 

Sun’s parallax, subtract . 6 

Correction to be subtracted, 21 4 

©’s observed altitude, . . 51 42 50 

G’s tree altitude, . . . 51° 21' 46" 


Declination Naut. Almanac, 7° 3'19" 
Correct. 45° W. long., add 2 48 


True declination, . . D 

= 7 

6 

7 

Add. 

90 




97 

6 

7 

True altitude, subtract A 

= 51 

21 

46 


The required latitude, l = 45° 44' 21" 


Artificial Horizon. 

When the observation is made by a sextant through an artificial horizon, the 
ol/served angle must be divided by 2 for the altitude, and there will be no correc¬ 
tion for dip of horizon, nor for semi-diameter, as the sun’s discs are brought to 
cover one another. When a regular qnicksilver horizon is not at hand, some 
slimy liquid, like oil or molasses, in an open vessel, may be used in calm weather. 

In perfectly calm weather the altitude may be taken in a pool of water, which 
has been done by the author in South America. 

























Astronomy. 


505 


Apparent, Time. 

Example. On the 8th of February, 1872, the sun’s true altitude was found to be 
a = 30° 46' 29" in the afternoon, the latitude Z = 38° 18'38" N., and declination 
corrected D =15° 5' 10" S. Required, the apparent and mean time of obser¬ 
vation ? 

The latitude and declinations are of different names. 

Cos. L = sin.30° 46' 29" X sec.3S° 18' 38" X sec.15 0 5' 10" = 0.68763 
+ tan.38° 18' 38" X tan,15° 5' 10" = 0.21293 
Apparent time of obs., L = 1 li. 49m. 13s. = cos.29° 18' 15" = 0.88856 
Equation of time, add 14 26 

Mean time, 2 h. 3m. 39s., the time required. 

The calculation with logarithms is set up as follows : 

Log. sin.30° 46' 29" = 9.7089S 
“ sec.38 18 38 =0.10531 Tan. =9.89765 

“ “ 15 5 10 = 0.01529 Tan. = 9.43067 

Logarithms, . . . 9.82958 9.32832 

Natural numbers, . 0.67563 0.2L293 

Add for different names, 0.21293 

App. time, L = 1 h. 49m. 13s. = 27° 18' 15" = 0.S8856 = cosino for hour angle. 

The hour angle in time can be read off directly for the cosine in the trigono¬ 
metrical tables. 

When the observation is made in the forenoon, subtract the cosine hour angle 
from 12 hours ; or it can be read off directly from the tables by calling cosine for 
sine L. 

To Find tlie Longitude. 

Small differences in longitude can be obtained from actual measurement, as 
explained in Plane Sailing and Traverse Surveying. 

! For great distances, it is necessary to know the simultaneous times at the two 
meridians between which the longitude is required. 

At sea, the time at a distant meridian is kept by a chronometer, generally regu¬ 
lated for Greenwich mean time, and the difference in time between the two merid¬ 
ians is the difference in longitude. 

The Greenwich mean time can also be found by astronomical, observations and 
the Nautical Almanac. The common astronomical observations for Greenwich 
mean time are the moon culminating stars, lunar distances, eclipses of the sun 
and, moon, all of which are too complicated to be introduced in this Pocket-book. 

Eclipses of Jupiter’s Satellites. 

The most simple astronomical observations for Greenwich mean time are of the 
eclipses of Jupiter’s satellites, but, unfortunately, the tables for those eclipses in 
the English Nautical Almanac are not yet reliable, as has been found by the author 
in using these tables in the interior of South America. The observation of several 
eclipses' in one locality did not agree with the time given in the Nautical Almanac, 
from which I suspected the tables to be incorrect. On my arrival in London, in 
August, 1871.1 called on the superintendent of the Nautical Almanac, Professor Hind, 
and explained my experience in relation to those eclipses and tables. The distin¬ 
guished professor very kindly informed me that “there has not yet been sufficient 
accurate observations made of the time of eclipses of Jupiter’s satellites to make the 
tables reliable—the actual time of eclipse may differ minutes from that in the tables.” 

The tables of eclipses of Jupiter’s satellites are omitted in the American Nauti¬ 
cal Almanac, perhaps for the same reason as given by Professor Hind. 

It is strange that the elements of Jupiter’s satellites are not yet better known ; 
there must be a screw loose somewhere. 

This is a subject well worthy of attention at the National Observatory at Wash¬ 
ington. The eclipses of Jupiter’s satellites can be observed by an ordinary good 
spy-glass, even at sea, in calm weather, which would be of great importance over 
the whole world if accurate tables could be procured. Chronometers could then 
be corrected with great precision to Greenwich time in any part of the world, and 
by simple addition or subtraction of time the longitude could be determined cor¬ 
rectly without the aid of complicated, bulky and expensive instruments, which 
few can afford to buy or know how to manage. 

There are some 300 eclipses in Jupiter’s satellites every year. 
















506 


Astronomy. 


Elements of the Planetary System. 


The 

tn 

Mean 

Sidereal 

Ecceut. 

Diam- 

Yel orbtl Rota- 




principal 


distance 

period, 

part, 

eter in 

Miles 

tion in 


Mass. 

Volume. 

Planets. 

55 

fr. Sun. 

Days. 

sm. axis 

miles. 

per sec. 

hours. 

ity. 



Sun,. . 

o 

. # 

# # 

. . 

882000 


H. M.» 
607 48 

0.25 

355000 

1378000 

Mercury, 

0 

0.3871 

87.969 

0.2055 

3140 

30.4 

24 05 

1.12 

0.06966 

0.06218 

Venus, 

? 

0.7233 

224.70 

0.0068 

7800 

22.3 

23 21 

0.92 

0.877 

0.9531 

Earth, . 

© 

1. 

365.25 

0.0168 

7912 

18.9 

24 0 

1. 

1. 

1. 

Mars, . 

c? 

1 5236 

e^oos 

0.0933 

4100 

15.33 

24 37 

0.95 

0.1313 

0.1384 

Jupiter, 


5.2028 

4332.6 

0.0482 

87000 

8.31 

9 56 

0.24 

317.5 

1322.5 

Saturn, 

h 

9.5388 

10759 

0.0561 

79160 

6.14 

10 29 

0. i 4 

139.5 

996.2 

Uranus, 


19.182 

30687 

0.0467 

34500 

4.33 

9 30 

0.21 

198. 

82.47 

Neptune, 


30.037 

60127 

0.0087 

41500 

3.45 

• • 

0.14 j 

20. 

143.5 


Position of some Stars of tlie 1st and 
3d Magnitudes, dan. 1, 1874, 

Name of Star. 


N. Hemisphere, 
a Andromeda;, 
a Polaris, 
a Arietis, 
a Persi, 
a Aldebaran, 
a Capella, 

0 Tauri, 
a 2 Castor, 
a Procyon, 

0 Pollux, 
a Regains, 
y 1 Leonis, 
a Great Bear, 
y Great Bear, 
7] Great Bear, 
a Arcturus, 
a Coronas, 

£ Herculis, 
a 1 Herculis, 
a Vega, 
a Altair, 

S. Hemisphere. 

0 Ceti, 
a Acbernar, 

0 Hi gel, 

8 Orionis, 
a Cauopus, 
a Sirius, 
e Canis Major, 
i Argus, 
a Hydra;, 
a 1 Crucis, 
a Spica, 

0 Centauri, 
a 1 Centauri, 

0 Libras, 

0 1 Scorpii, 
a An tares, 
a Australis, 
a Pavonis, 

0 Aquarii, 
a Cruis, 
a Fomalhaut, 


Mg. 

Right 

Ann. 

Decli- 

Ann. 

td. 

As’sion. 

Var. 

nation. 

1 

ar. 







North. 




it 

m. 

s. 


S. 

O 

' 

" 


" 

2 

0 

i 

52 

+ 

3.09 

28 

23 

41 

+ 

19.9 

2 

1 

12 

38 

+ 

20.7 

S8 

28 

15 


19.0 

2 

2 

0 

4 

+ 

3.36 

22 

51 

55 

+ 

17.2 

2 

3 

15 

20 

+ 

4.25 

49 

24 

38 

+ 

13.1 

1 

4 

28 

41 

+ 

3.43 

16 

15 

14 

+ 

7.61 

1 

5 

7 

23 

+ 

4.42 

45 

52 

1 

+ 

4.13 

2 

5 

18 

20 

+ 

3.79 

28 

29 

54 

+ 

3.42 

2.1 

7 

26 

33 

+ 

3.84 

32 

9 

45 


7.48 

1 

7 

32 

42 

4- 

3 14 5 

32 

45 


8.97 

1.2 

7 

37 

36 

+ 

3.68 

28 

19 

42 

_ 

8.35 

1.2 

10 

1 

40 

+ 

3.20 

12 

34 

56 

— 

17.4 

2 

10 

13 

l 

+ 

3.31 

20 

28 

41 


18.0 

2 

10 

55 

56 

+ 

3.76 

62 

25 

50 

_ 

19.4 

2.3 

11 

47 

12 

+ 

3.19 

54 

23 

42 

_ 

20.0 

2 

13 

42 

34 

+ 

2.37 

*49 

56 

33 

_ 

18.1 

1 

14 

9 

55 

+ 

2.73 

19 

50 

22 

_ 

18.S 

2 

15 

29 

21 

+ 

2 54 

27 

8 

24 

_ 

12.3 

3.2 

16 

36 

32 

+ 

2.26 

31 

49 

57 

_ 

6.69 

V ar. 

17 

8 

54 

+ 

2 73 

14 

32 

8 

_ 

4.39 

1 

18 

32 

40 

+ 

2.03 

38 

40 

3 

+ 

3.13 

1.2 

19 

44 

38 

+ 

2.93 

8 

32 

13 

”t“ 

9.22 







South. 



2 

0 

37 

16 

+ 

3.01 

18 

40 

43 

+ 

19.8 

1 

1 

33 

01 

+ 

2.23 

57 

52 

38 

+ 

18.4 

1 

5 

8 

29 

+ 

2.88 

8 

20 

57 

+ 

4.45 

2 

5 

25 

34 

+ 

3.06 

0 

23 

40 

+ 

2.96 

1 

6 

21 

9 

+ 

1.33 

52 

37 

39 


1.85 

1 

6 

39 

36 

+ 

2.64 

16 

32 

41 

— 

4.69 

2.1 

6 

53 

40 

+ 

2.36 

28 

48 

7 

— 

4.67 

2 

9 

13 

43 

+ 

1.60 

58 

44 

47 

— 

14.9 

2 

•9 

21 

24 

+ 

2.94 

8 

6 

49 

— 

15.4 

1 

12 

19 

36 

+ 

3.27 

62 

23 

58 

— 

19.9 

1 

13 

18 

33 

+ 

3.15 

10 

30 

11 

— 

18.9 

1 

13 

54 

57 

+ 

4.16 

59 

45 

50 

_ 

17.6 

1 

14 

31 

4 

+ 

4.04 

60 

18 

39 

_ 

15.0 

2 

15 

10 

14 

+ 

3.22 

8 

54 

59 

— 

13.5 

2 

15 

58 

7 

+ 

3.48 

19 

27 

31 

_ 

10.2 

1.2 

16 

21 

41 


3.67126 

9 

1 

— 

8.37 

2 

16 

35 

21 

+ 

6.28 

68 

47 

33 

— 

7.32 

2 

20 

15 

40 

+ 

4.79 

57 

8 

9 

+ 

111 

3 

21 

24 

55 

+ 

3 16 

6 

7 

28 

+ 

15.6 

2 

22 

0 

17 

+ 

3.81 

47 

34 

11 

“f" 

17.2 

1.2 |22 

50 

41 

+ 

3.331 

30 

17 

23 

+ 

19.0 


Tide Table.! 

Albany, N. Y., — 
Altona, Ger., + 
Amboy, N. Y., + 
Antwerp, Bel., + 
Baltimore, — 
Belfast, Ire., + 
Bergen, Nor., — 
Bordeaux, Fr., + 
Boston, Mass., + 
Boulogne, Fr., + 
Buenos Ayres, + 
Bremeu, + 

Cadiz, Spain, — 
Calais, Fr., + 
Calc’tta.Beng. + 
Charleston, + 
Cherbourg,Fr. + 
CapeHenry,A. + 

Cape G. Hope, j -|- 
Cape Horn, 1+ 

C. Hen!open, 

Dublin Bar, + 
Gibraltar, Sp., -j- 
Glasgow, Scot 
Hamburg, + 
llalifax, N. A., + 
Havana, Cuba, + 
Havre, I'r., + 

Hull, Eng., +4 22 
Key West,TJ.S. 

Lisbon B. Port 
Liverpool, 

New York, 

New Raven,A 
Newcastle, E. 

Norfolk, O.S., 

Ostend, Belg., 

Panama, 

Philadelphia, 
Portsmouth, 

Eng. & U. S„ + 
Piovidence, + 
Quebec, Can., 
j Queenstown, + 

Rio Janeiro, + 
Rotterdam, + 
Sandy Hook, + 
Valparaiso, -j- 
San Francisco, + 
i Washington, + 


h. m. 
0 40 

3 12 

5 27 
2 18 

1 52 

8 36 
0 37 

4 34 

9 20 
9 18 

6 6 

8 43 
0 22 

9 42 
0 23 

5 44 

8 23 
7 3 
0 29 

2 9 

6 41 

9 5 
0 13 
0 42 

3 22 
5 42 

7 41 
7 44 


+ 

+ 

+ 

+ 

+ 

+ 

+ 


7 22 
0 13 
9 16 
6 6 

8 38 
2 16 
5 14 

+10 18 
1 24 
— 0 49 


9 28 
5 25 
4 31 
2 54 
0 53 
1 38 
4 58 

7 2 

8 27 
1 58 


Rise 

feet 


12 


12 

19 


19 

5 

20 

4 

9 

5 

12 


8 

3 

12 

18 

2 

25 

5 

17 

7 

16 

24 

6 

10 

17 

6 

6 

5 

6 





















































Astronomy. 


507 


To Find the Meridian or True North by the North Star, 

Polaris. 


Pofaris ig not in the true north, hut revolves in 
co-declination on the 1st of January, 1872, which 
diminishes 19" every year; that on the 1st of Janu¬ 
ary, 1873, its co-declination will be 1° 22' 5". 

The position of Polaris is generally traced by the 
direction of the stars a and /3, in the Great Bear, 
which point nearly to the North Star. See figure. 

Polaris passes the meridian, or is true north when 
the star e in the Great Bear is perpendicular either 
over or under Polaris. In low latitudes the Polaris 
is near the horizon, and the star e cannot be seen 
when under, but must be observed at its upper 
transit. When the star e is horizontal with Polaris, 
substract the radius of the circle, and the remainder 
will be the true north, from which the variation of 
the compass is ascertained. There is no star near 
the South Pole from which a similar observation can 
be made. 


a circle of radius 1° 22' 24" 

^Polaris. 

I 1 \ 1 

1 I \ 


r 


* 

IV 


Table Showing how Much Earlier any Fixed Star Passes 

the Meridian , rises or.sets, in number of days or nights up to 100. 


CO 

+-• 

rP 



CO 

rC 


GO 

4—' 



GO 

4-^ 

A 



GO 

-M 


bn 

2 

H.M. S. 


on 

£ 

H.M. S. 

.bn 

2 

H.M. S. 


• SP 

2 

H.M. S. 


'bn 

2 

H.M. S. 

i 

0 3 55.9 


11 

0 43 15.0 

21 

1 22 34.1 


31 

2 01 53.2 


45 

2 56 55.0 

2 

0 7 51.8 


12 

0 47 10.9 

22 

1 26 30.0 


32 

2 05 49.1 


50 

3 16 35.9 

3 

0 11 47.7 


13 

0 51 06.8 

23 

1 30 25.9 


33 

■2 09 45 ;0 


55 

3 36 15.0 

4 

0 15 43.6 


14 

0 55 02.7 

24 

1 34 21.8 


34 

2 13 40.9 


60 

3 55 54.5 

5 

0 19 39.5 


15 

0 58 58.6 

25 

1 38 17.7 


35 

2 17 36.8 


65 

4 15 34.1 

6 

0 23 35.5 


16 

1 02 54.5 

26 

1 42 13.6 


36 

2 21 32.7 


70 

4 35 13.6 

7 

0 27 31.4 


17 

1 06 50.5 

27 

1 46 09.6 


37 

2 25 28.6 


75 

4 54 53.2 

8 

0 31 27.3 


18 

110 46.4 

28 

1 50 05.5 


38 

2 29 24.5 


80 

5 14 32.7 

9 

0 35 23.2 


19 

114 42.3 

29 

1 54 01.4 


39 

2 33 20.4 


90 

5 53 51.8 

10 

0 39 19.1 


20 

1 18 38.2 

30 

1 57 57.3 


40 

2 37 16.4 


100 

6 33 10.9 


The preceding table is for regulating a watch, clock or chronometer. The fixed 
stars set 3 m. 55.909 s. earlier every day, and by observing the time of setting 
over a sharp, distant object, as a hill, mountain or a house, the time-keeper can be 
regulated with great precision. 

Example. A fixed star is observed to set at 9 h. 35 rn. 51 s. 

Twenty-five days after, the same star set at 7 h. 66 m. 49 s. 

Add the correction for 25 days, ... 1 38 17.7 

Sum,.~9 35 067 

The time-keeper has lost 51 — 06.7 = 44.3 seconds in 25 days, or 1.772 seconds per 
day. 


To Find the Time When Any Star or Planet Passes the 

Meridian. 

Subtract the sun’s right ascension from that of the star, increased by 24 vf neces¬ 
sary, and the remainder will be the apparent time when the star passes the 
meridian. The sun’s R. A. must be corrected for longitude from Greenwich, and 
for time of observation. This is the best mode of finding the meridian and varia¬ 
tion of the compass, but the apparent time must be correctly known. 

To Find 'Which Star Passes the Meridian Near a Desired Time. 

Add the sun’s R. A. to the desired hour, and the sum will be the nearest R. A. 
of the star passing the meridian at that time. Reject 24 hours if necessary. Find 
in the table of stars the one which comes nearest to that R. A. 




























SOS 


High Water. 


TO APPROXIMATE THE TIME OF HIGH WATER. 

On account of the Moon’s orbit being an ellipse, in which the Earth is one of the 
focuses, and that the major axis of that ellipse does not point to the Sun, but to a 
fixed constellation of stars, the actual time of high water and also that of passing 
the meridian are not equal for equal age of the Moon, but may differ as much as 15 
minutes from the average in the accompanying table. Also, the force and direc¬ 
tion of winds cause a still greater variation. 

Find first the Moon’s age for the given day, as described on page 496. Opposite 
the age in the table is the time of the day when the Moon is south, or passes the 
meridian, and in the following columns are the times of high water at London 
Bridge in the morning and afternoon. Add or subtract the time in the tide table, 
page 506, for any other location, and the sum or difference is the time of high water 
at that place. 



Moon 

• 



High water at 
London Bridge. 

Quart’r 

Face 

Age 

d. 

h. 

South. 

m. 

Age. 

Morning, 
h. m. 

Afternoon 
h. xu. 

New. 

(V) 

0 

12 

0p 

. m. 

0 

1 

59 

2 

7 

M 


1 

12 

49 

66 

1 

2 

21 

2 

36 

e-t- 

►o 

._£ 

© 

2 

3 

1 

2 

38 

26 

u 

u 

2 

3 

2 

3 

50 

18 

3 

3 

3 

33- 

p 


4 

3 

26 

a 

4 

3 

47 

4 

2 

CD 

Q 

5 

4 

4 

66 

5 

4 

16 

4 

31 


\s ty 

6 

4 

55 

66 

6 

4 

50 

5 

5 

Half. 

© 

7 

8 

6 

6 

42 

30 

(6 

66 

7 

8 

5 

6 

24 

7 

5 

6 

45 

35 

to 


9 

7 

19 

66 

9 

7 

7 

7 

46 

►o 


10 

8 

8 

CC 

10 

8 

33 

9 

25 

r* 


11 

8 

57 

66 

11 

10 

14 

10 

54 


/jgjlBjK 

12 

9 

46 

(( 

12 

11 

28 

11 

54 

CD 

llipf 

13 

10 

34 

66 

13 

Noon. 

0 

17 

* 


14 

11 

23 

66 

14 

0 

40 

1 

0 

Full. 

f§ 

15 

12 

12 a 

. m 

15 

1 

20 

1 

40 



16 

1 

1 

66 

16 

2 

1 

2 

22 

Cu 


17 

1 

50 

66 

17 

2 

42 

3 

5 

►o 

fsjy 

18 

2 

38 

66 

18 

3 

26 

3 

48 

P 


19 

3 

27 

66 

19 

4 

10 

4 

34 

ct- 


20 

4 

16 

66 

20 

4 

55 

5 

19 



21 

5 

5 

66 

21 

5 

44 

6 

12 

Half. 

© 

22 

23 

5 

6 

54 

42 

66 

66 

22 

23 

6 

7 

44 

59 

7 

8 

21 

43 


/jpk. 

24 

7 

31 

66 

24 

9 

31 

10 

15 

fj 

25 

8 

20 

66 

25 

10 

52 

11 

23 



26 

9 

9 

66 

26 

11 

49 

M’night. 


( 0 ) 

27 

9 

58 

66 

27 

0 

11 

0 

29 



28 

10 

46 

66 

28 

0 

48 

1 

5 

CD 


29 

11 

35 

66 

29 

1 

23 

1 

36 


© 

29 5 

12 


66 

29i 

1 

51 

2 

7 


High. Water. 

Examples. 

Required, the time of 
H. W. in Philadelphia on 
the 8th of Feb., 1874? 

Epact, year, . 12 pg. 497. 
Epact, month, 2pg. 496. 
Date, February, 8 • 

Moon’s age, . 22 days. 
H.W.Lond., 6h.44m. 

Phila. subt., 0 49 pg. 506. 


H.W.Phila., 5h.55m. 
in the morn., Feb. 8,1874. 

Required, the time of 
H. W. in Panama on the 
7th of October, 1873? 

Epact, year, 1 pg. 497. 
Epact. month, 8 pg. 496. 
Date in October, 7 

Moon’s age, 16 days. 

II.W. Lond., 2h. lm. a.m. 
Panama add 1 24, pg. 506 

II.W. 3h. 25m. a.m. 

in Panama. 


Elements of Jupiter’s Satellites. 

Order 

of 

satellite. 

1st. 

2d. 

3d. 

4th. 

Radius of 
orbit, that 
of Jupiter=l 
6,0-185 
9.6235 
15.3502 
26.99!- 3 

Time in days 
of one 
revolution. 
1.7691 
5.5512 
7.1546 
16.6888 

Revolutions 
ar. Jupiter 
per vear. 
206.457 
65.7952 
51.0499 
21.8855 

Mass, 
that of 
Jupiter=l 
0.000017 
0.000028 
0.000088 
0.000043 

Diameter 
of satellite 
miles. 



A’umber of Moons or Satellites to each Planet. 

Earth, 1. Jupiter, 4. Saturn, with rings, 8. Uranus, 8 moons. 













































Soundings, 


509 


SOUNDINGS. 


To Reduce Soundings to Low Water. 

Letters denote — 

T= time in hours between high and low water. 

t = time in hours from low water to the time when the soundings are taken. 
R = vertical rise of tide in feet from high to low water, 
r = reduction of the sounding taken at the time, t, in feet. 

v ~—~, and r = \R (1 =f cos.v), 

— cos.v when v <[ 90 
-f cos.v when v > 90 

Example. High water at 10 h. 15 m. p. m. 

Low water at 3 h. 45 m. “ 

Time T= 6 li. 30 m. “ 

The sounding taken at 5 h. 30 m. “ was 16 feet 6 inches. 
Time t = 1 h. 45 m. 

Vertical rise R = 9.75 feet. 

Required, the reduction r — ? and true sounding at low water? 

v = 18 - 0 x - 1 -- = 48° 30', cos.v = 0.66262. 

6.5 

Reduction r — £ X 9-75 (1 — 0.66262) — 1.6445 feet. 

Sounding taken at 5 h. 30 m. was 16.5 feet. 

Reduction'subtract r = 1.6447 

True sounding at low water, 14.8553 feet. 

Reduction for Soundings to L<ow Water. 

This table will answer for any unit of measure of rise. 


Rise 

Time of soundin. 

; in ins. and min. 

before or after that of high water. 

Rise 

R. 

0.30 

1 

1.30 

2 

2.30 

3 

3.30 

4 

4.30 

5 

5.30 

6 

R. 

1 

0.98 

0.94 

0.87 

0.78 

0.67 

0.55 

0.43 

0.31 

0.20 

0.12 

0.05 

0.01 

1 

2 

1.97 

1.88 

1.74 

1.56 

1.34 

1.10 

0.86 

0.62 

0.40 

0.24 

0.10 

0.02 

2 

3 

2.95 

2.82 

2.61 

2.34 

2.01 

1 65 

1.29 

0.93 

0.60 

0.36 

0.15 

0.03 

3 

4 

3.93 

3.76 

3.48 

3.12 

2.68 

2.20 

1.72 

1.25 

0.82 

0.46 

0.20 

0.04 

4 

5 

4.92 

4.70 

4.35 

3.90 

3.35 

2.74 

2.15 

1.56 

1.03 

0.58 

0.25 

0-05 

5 

6 

5.91 

5.65 

5.22 

4.68 

4.03 

3.30 

2.58 

1.87 

1.23 

0.69 

0.30 

0.06 

6 

7 

6.90 

6.59 

6 10 

5.46 

4.70 

3.84 

3.01 

2.18 

1.44 

0.S1 

0.35 

0.07 

7 

8 

7.88 

7.52 

6.97 

6.24 

5.36 

4.40 

3.44 

2.50 

1.65 

0 93 

0.40 

0.08 

8 

9 

8.86 

8.47 

7.84 

7.02 

6.03 

4.94 

3.87 

2.80 

1.85 

1.04 

0.45 

0.09 

9 

10 

9.85 

9.41 

8.71 

7.79 

6.71 

5.52 

4.30 

3.12 

2.06 

1.16 

0.50 

0.10 

10 

11 

10.9 

10.3 

9.59 

8.59 

7.39 

605 

4.74 

3.43 

2.27 

1.28 

0.55 

0.11 

11 

12 

11.9 

11.3 

10.5 

9.37 

8.06 

6.60 

5.16 

3.74 

2.47 

1.40 

0.60 

0.12 

12 

13 

12.8 

12.2 

11.3 

10.1 

8.72 

7.14 

5.60 

4.05 

2.63 

1.51 

0.65 

0.13 

13 

14 

13.8 

13.2 

12.2 

11.0 

9.40 

7.70 

6.02 

3.36 

2.89 

1.62 

0.70 

0.14 

14 

15 

14.8 

14.1 

13.0 

117 

10.0 

7.25 

6.45 

3.67 

3.09 

1.74 

0.75 

0.15 

15 

16 

15.8 

15.0 

14.0 

12.5 

10.7 

8.78 

6.88 

5 .GO 

3.30 

1.85 

o.so 

0.16 

16 

17 

16.8 

16.0 

14.8 

13.3 

11.4 

9.35 

7.31 

5.25 

3.50 

1 97 

0.85 

0.17 

17 

18 

17.8 

17.0 

15.7 

14.0 

12.1 

9.90 

7.75 

5.60 

370 

204 

0.90 

0.18 

18 

19 

18 7 

17.9 

16.6 

14.8 

12.8 

10.4 

8.17 

5.91 

3.81 

2.20 

0.95 

0.19 

19 

20 

19.7 

18.9 

17.5 

15.6 

134 

11.0 

8.60 

6.23 

4.11 

2.32 

1.00 

0.20 

20 

21 

20.7 

19.8 

18.3 

16.4 

14.1 

11.5 

9.04 

6.54 

4.32 

2.43 

1.05 

0.21 

21 

22 

217 

20.7 

. 19.2 

17.2 

14 8 

12.1 

9.16 

6.85 

4.53 

2.55 

1.10 

0.22 

22 

23 

22.7 

21.7 

20.0 

18.0 

15.4 

12.6 

9.90 

7.16 

4.73 

2.67 

1.15 

0.23 

23 

24 

23.7 

22.6 

20.9 

18.7 

16.1 

13.2 

10.3 

7.47 

4.94 

2.78 

1.20 

0.24 

21- 

25 

21.7 

23.5 

21.8 

19.5 

16.8 

13.7 

10 s 

7 78 

5.14 

2.90 

1.25 

0.25 

25 

26 

25.6 

21.5 

22 7 

2 1.3 

17.4 

14.3 

11.2 

8.10 

5.52 

3.01 

1.30 

0.26 

26 

27 

26.6 

25.4 

23.5 

21.1 

18.1 

14.9 

11.6 

8.41 

5.55 

3.13 

1.35 

0.27 

27 

28 

27.6 

26.4 

21.4 

21.9 

18.S 

15.4 

12.0 

8.72 

5.76 

3.25 

1.40 

0.28 

28 

29 

28 6 

27 3 

25.3 

22.7 

19.4 

16.0 

12.5 

9.03 

5.96 

3.36 

1.45 

0 29 

29 

30 

29.6- 

28.3 

26.2 

23.4 

20.1 

16.5 

12.9 

934 

6.17 

3.48 

1.50 

0.30 

30 

R. 

5.45 

5.15 

4.45 

4.15 

3.45 

3.15 

2.45 

2.15 

1.45 

1.15 

0.45 

0.15 

R. 

Rise 

1 irne of sounding in hrs. and min. before or after that of low water. 

Rise 


43 * 
















































510 


Astronomy. 


To Find at wliat Time tlie Sun Sets and Rises. 

Let v denote the time angle from 6 o’clock to when the sun sets or rises, then— 

Sin.w = tan.? tan.D. 

Example. What time does the sun set and rise on the 2lst of June, in 60° lati¬ 
tude? 

The declination on this day is about 23° 27'. 

Sin.v = tan.60° X tau.23° 27' = 0.75131 = sin.37i. 14 m. 48s. 

The sun rises at 2 h. 45m,. 12s., and sets at 9 h. 14m. 48s. 

To Find tlie Length of Day and Night* 

Day.—D ouble the time of sunset, is the length of the day. 

Night. —Double the time of sunrise, is the length of the night. 

Amplitude. 

The angle or bearing from east or west to where any heavenly body sets or rises, 
is called the amplitude of that body, which, denoted by x, will be— 

Sin.a; = sec.? sin.Z>. 

The amplitude is used for finding the variation of the compass. 

Example. The sun’s declination being 18° 25' south, required, his amplitude in 
latitude 48° 45' north ? 

Sin.a: = sec.48° 45' X sin.l8° 25' = sin.28° 38' south, 
the amplitude required. 

Azi mutli. 

I — latitude, D — declination, and a — altitude. 
z = angle of azimuth, or bearing of the heavenly body from meridian to the 
pole above horizon. 

When the latitude and declination are of 


Equal Names— 

l+a — D + 90 

m =- 


n = 


m 


^ 90 -!>)• 


Different Names— 

l+a+D + 90 


m 


2 


Subtract the smallest, and the 
remainder is n. 


Cos. ^ z = |/ cos.m cos.n sec .1 sec .a. 













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